stabilized finite element methods for convection-diffusion

Stabilized Finite Element Methods

for Convection-Diffusion-Reaction, Helmholtz and Stokes Problems

P. Nadukandi E. Oñate

J. García-Espinosa

Monograph CIMNE Nº-130, July 2012

Stabilized Finite Element Methods for Convection-Diffusion-Reaction,

Helmholtz and Stokes Problems

P. Nadukandi E. Oñate

J. García-Espinosa

Monograph CIMNE Nº-130, July 2012

International Center for Numerical Methods in Engineering Gran Capitán s/n, 08034 Barcelona, Spain

INTERNATIONAL CENTER FOR NUMERICAL METHODS IN ENGINEERING Edificio C1, Campus Norte UPC Gran Capitán s/n 08034 Barcelona, Spain www.cimne.com First edition: March 2012 HIGH-PERFORMANCE MODEL REDUCTION PROCEDURES IN MULTISCALE SIMULATIONS Monograph CIMNE M127 The authors ISBN: 978-84-940243-3-7 Depósito legal: B-23420-2012

A B S T R A C T

We present three new stabilized finite element (FE) based Petrov–Galerkin methodsfor the convection–diffusion–reaction (CDR), the Helmholtz and the Stokes problems,respectively. The work embarks upon a priori analysis of some consistency recoveryprocedures for some stabilization methods belonging to the Petrov–Galerkin frame-work. It was found that the use of some standard practices (e. g.M-Matrices theory)for the design of essentially non-oscillatory numerical methods is not appropriatewhen consistency recovery methods are employed. Hence, with respect to convectivestabilization, such recovery methods are not preferred. Next, we present the design ofa high-resolution Petrov–Galerkin (HRPG) method for the CDR problem. The struc-ture of the method in 1d is identical to the consistent approximate upwind (CAU)Petrov–Galerkin method [68] except for the definitions of the stabilization parameters.Such a structure may also be attained via the Finite Calculus (FIC) procedure [141] byan appropriate definition of the characteristic length. The prefix high-resolution is usedhere in the sense popularized by Harten, i. e.second order accuracy for smooth/regu-lar regimes and good shock-capturing in non-regular regimes. The design procedurein 1d embarks on the problem of circumventing the Gibbs phenomenon observedin L2 projections. Next, we study the conditions on the stabilization parameters tocircumvent the global oscillations due to the convective term. A conjuncture of thetwo results is made to deal with the problem at hand that is usually plagued byGibbs, global and dispersive oscillations in the numerical solution. A multi dimen-sional extension of the HRPG method using multi-linear block finite elements is alsopresented.

Next, we propose a higher-order compact scheme (involving two parameters) onstructured meshes for the Helmholtz equation. Making the parameters equal, we re-cover the alpha-interpolation of the Galerkin finite element method (FEM) and theclassical central finite difference method. In 1d this scheme is identical to the alpha-interpolation method [140] and in 2d choosing the value 0.5 for both the parame-ters, we recover the generalized fourth-order compact Padé approximation [81, 168](therein using the parameter γ = 2). We follow [10] for the analysis of this scheme andits performance on square meshes is compared with that of the quasi-stabilized FEM[10]. Generic expressions for the parameters are given that guarantees a dispersion ac-curacy of sixth-order should the parameters be distinct and fourth-order should theybe equal. In the later case, an expression for the parameter is given that minimizes themaximum relative phase error in 2d. A Petrov–Galerkin formulation that yields theaforesaid scheme on structured meshes is also presented. Convergence studies of theerror in the L2 norm, the H1 semi-norm and the l∞ Euclidean norm is done and thepollution effect is found to be small.

Finally, we present a collection of stabilized FE methods derived via first-order andsecond-order FIC procedures for the Stokes problem. It is shown that several wellknown existing stabilized FE methods such as the penalty technique, the GalerkinLeast Square (GLS) method [93], the Pressure Gradient Projection (PGP) method[35] and the orthogonal sub-scales (OSS) method [34] are recovered from the general

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residual-based FIC stabilized form. A new family of Pressure Laplacian Stabilization(PLS) FE methods with consistent nonlinear forms of the stabilization parameters arederived. The distinct feature of the family of PLS methods is that they are nonlin-ear and residual-based, i. e.the stabilization terms depend on the discrete residualsof the momentum and/or the incompressibility equations. The advantages and dis-advantages of these stabilization techniques are discussed and several examples ofapplication are presented.

Keywords : Convection–diffusion–reaction problem, Helmholtz equation, Stokesflow, Stabilized finite element methods, High-resolution Petrov–Galerkin method, lin-ear interpolation of FEM and FDM stencils, Pressure Laplacian Stabilization, Disper-sion analysis.

R E S U M E N

Presentamos tres nuevos métodos estabilizados de tipo Petrov–Galerkin basado enelementos finitos (FE) para los problemas de convección–difusión–reacción (CDR), deHelmholtz y de Stokes, respectivamente. El trabajo comienza con un análisis a prioride un método de recuperación de la consistencia de algunos métodos de estabilizaciónque pertenecen al marco de Petrov–Galerkin. Hallamos que el uso de algunas de lasprácticas estándar (por ejemplo, la teoría de Matriz-M) para el diseño de métodosnuméricos esencialmente no oscilatorios no es apropiado cuando utilizamos los méto-dos de recuperación de la consistencia. Por lo tanto, con respecto a la estabilizaciónde convección, no preferimos tales métodos de recuperación. A continuación, presen-tamos el diseño de un método de Petrov–Galerkin de alta-resolución (HRPG) parael problema CDR. La estructura del método en 1d es idéntico al método CAU [68]excepto en la definición de los parámetros de estabilización. Esta estructura tambiénse puede obtener a través de la formulación del cálculo finito (FIC) [141] usando unadefinición adecuada de la longitud característica. El prefijo de alta-resolución se uti-liza aquí en el sentido popularizado por Harten, es decir, tener una solución con unaprecisión de segundo orden en los regímenes suaves y ser esencialmente no oscila-toria en los regímenes no regulares. El diseño en 1d se embarca en el problema deeludir el fenómeno de Gibbs observado en las proyecciones de tipo L2. A continuación,estudiamos las condiciones de los parámetros de estabilización para evitar las oscila-ciones globales debido al término convectivo. Combinamos los dos resultados (unaconjetura) para tratar el problema CDR, cuya solución numérica sufre de oscilacionesnuméricas del tipo global, Gibbs y dispersiva. También presentamos una extensiónmultidimensional del método HRPG utilizando los elementos finitos multi-lineales.

A continuación, proponemos un esquema compacto de orden superior (que incluyedos parámetros) en mallas estructuradas para la ecuación de Helmholtz. Haciendoigual ambos parámetros, se recupera la interpolación lineal del método de elementosfinitos (FEM) de tipo Galerkin y el clásico método de diferencias finitas centradas. En1d este esquema es idéntico al método AIM [140] y en 2d eligiendo el valor de 0.5 paraambos parámetros, se recupera el esquema compacto de cuarto orden de Padé gener-alizada en [81, 168] (con el parámetro γ = 2). Seguimos [10] para el análisis de este

iv

esquema y comparamos su rendimiento en las mallas uniformes con el de FEM cuasi-estabilizado (QSFEM) [10]. Presentamos expresiones genéricas de los parámetros quegarantiza una precisión dispersiva de sexto orden si ambos parámetros son distintos yde cuarto orden en caso de ser iguales. En este último caso, presentamos la expresióndel parámetro que minimiza el error máximo de fase relativa en 2d. También pro-ponemos una formulación de tipo Petrov–Galerkin que recupera los esquemas antesmencionados en mallas estructuradas. Presentamos estudios de convergencia del er-ror en la norma de tipo L2, la semi-norma de tipo H1 y la norma Euclidiana tipo l∞ ymostramos que la pérdida de estabilidad del operador de Helmholtz (pollution effect)es incluso pequeña para grandes números de onda.

Por último, presentamos una colección de métodos FE estabilizado para el prob-lema de Stokes desarrollados a través del método FIC de primer orden y de segundoorden. Mostramos que varios métodos FE de estabilización existentes y conocidoscomo el método de penalización, el método de Galerkin de mínimos cuadrados (GLS)[93], el método PGP (estabilizado a través de la proyección del gradiente de presión)[35] y el método OSS (estabilizado a través de las sub-escalas ortogonales) [34] se re-cuperan del marco general de FIC. Desarrollamos una nueva familia de métodos FE,en adelante denominado como PLS (estabilizado a través del Laplaciano de presión)con las formas no lineales y consistentes de los parámetros de estabilización. Una car-acterística distintiva de la familia de los métodos PLS es que son no lineales y basadosen el residuo, es decir, los términos de estabilización dependerá de los residuos dis-cretos del momento y/o las ecuaciones de incompresibilidad. Discutimos las ventajasy desventajas de estas técnicas de estabilización y presentamos varios ejemplos deaplicación.

Palabras clave : El problema de convección–difusión–reacción, La ecuación deHelmholtz, El problema de Stokes, Métodos de elementos finitos estabilizado, Elmétodo de Petrov–Galerkin con alta-resolución, Interpolación lineal de las esténcilesdel MEF y del MDF, Cálculo finito, Análisis de dispersión numérica.

v

Every morning in Africa, a gazelle wakes up.It knows it must run faster than the fastest lion or it will be killed.

Every morning in Africa, a lion wakes up.It knows it must outrun the slowest gazelle or it will starve to death.

It doesn’t matter whether you are a lion or a gazelle;when the sun comes up, you’d better be running.

— Herbert Eugene Caen.

P U B L I C AT I O N S

This monograph contains some unpublished material, but is mainly based on thefollowing publications wherein some ideas and figures have appeared previously.

I Prashanth Nadukandi, Eugenio Oñate, and Julio García. Analysis of a consis-tency recovery method for the 1d convection–diffusion equation using linearfinite elements. International Journal for Numerical Methods in Fluids, 57(9):1291–1320, July 2008. ISSN 02712091. doi: 10.1002/fld.1863.

II Prashanth Nadukandi, Eugenio Oñate, and Julio García. A high-resolution Petrov–Galerkin method for the 1d convection–diffusion–reaction problem. ComputerMethods in Applied Mechanics and Engineering, 199(9–12):525–546, January 2010.ISSN 00457825. doi: 10.1016/j.cma.2009.10.009.

III Prashanth Nadukandi, Eugenio Oñate, and Julio García. A fourth-order compactscheme for the Helmholtz equation: Alpha-interpolation of FEM and FDM sten-cils. International Journal for Numerical Methods in Engineering, 86(1):18–46, April2011. ISSN 00295981. doi: 10.1002/nme.3043.

IV Eugenio Oñate, Prashanth Nadukandi, Sergio R. Idelsohn, Julio García and Car-los A. Felippa. A family of residual-based stabilized finite element methods forStokes flows. International Journal for Numerical Methods in Fluids, 65(1–3):106–134,January 2011. ISSN 02712091. doi: 10.1002/fld.2468.

V Prashanth Nadukandi, Eugenio Oñate, and Julio García. A Petrov–Galerkin for-mulation for the alpha-interpolation of FEM and FDM stencils. Applications tothe Helmholtz equation. International Journal for Numerical Methods in Engineer-ing, 89(11):1367–1391, March 2012. ISSN 00295981. doi: 10.1002/nme.3291.

VI Prashanth Nadukandi, Eugenio Oñate, and Julio García. A high-resolution Petrov–Galerkin method for the convection–diffusion–reaction problem. Part II—A mul-tidimensional extension. Computer Methods in Applied Mechanics and Engineering,213–216:327–352, March 2012. ISSN 00457825. doi: 10.1016/j.cma.2011.10.003.

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http://doi.wiley.com/10.1002/fld.1863

http://linkinghub.elsevier.com/retrieve/pii/S0045782509003545

http://doi.wiley.com/10.1002/nme.3043



http://www.sciencedirect.com/science/article/pii/S004578251100315X

When eating bamboo sprouts, remember the man who planted them

— a Chinese Proverb

A C K N O W L E D G M E N T S

The authors thanks Profs. Ramon Codina, Sergio Idelsohn, Carlos Felippa, AssadOberai for many useful discussions.

The first author acknowledges the economic support received through the FI pre-doctoral grant (ref. 2006FI 00836) from the Department of Universities, Research andInformation Society (Generalitat de Catalunya) and the European Social Fund.

This work was partially supported by the SAFECON project of the European Re-search Council (European Commission).

ix

C O N T E N T S

introduction 1

i convection-diffusion-reaction problem 5

1 analysis of a consistency recovery method 7

1.1 Introduction 7

1.2 Transient Convection Diffusion Equation 8

1.2.1 Problem Statement 8

1.2.2 Dispersion Relation in 1d 10

1.3 FE Discretization 10

1.3.1 Semi-Discrete Form 10

1.3.2 DDR in 1d 12

1.3.3 DDR Plots 14

1.3.4 Discussion 15

1.4 Stabilization Parameters 25

1.5 Discrete Maximum Principle 29

1.6 Numerical Examples 29

1.6.1 Example 1 29

1.6.2 Example 2 30

1.6.3 Example 3 33

1.6.4 Example 4 33

1.7 Conclusions 36

2 a high-resolution petrov–galerkin method in 1d 37

2.1 Introduction 37

2.2 High-resolution Petrov–Galerkin method 39

2.3 Derivation of the HRPG expression via the FIC procedure 42

2.4 Gibbs phenomenon in L2 projections 43

2.4.1 Introduction 43

2.4.2 Galerkin Method 44

2.4.3 HRPG design 46

2.4.4 Examples 48

2.4.5 Summary 53

2.5 Convection–diffusion–reaction problem 53

2.5.1 Galerkin method and discrete upwinding 53

2.5.2 Model problem 2 54




2.5.6 Summary 59

2.5.7 Examples 59

2.6 Conclusions 66

3 multidimensional extension of the hrpg method 69

3.1 Introduction 69

xi

xii contents

3.2 Quantifying characteristic layers 70

3.3 Objective characteristic tensors 74

3.4 Stabilization parameters 76

3.5 Examples 77

3.5.1 Steady-state examples 77

3.5.2 Transient examples 82

3.5.3 Discussion 85

3.6 Conclusions 91

ii helmholtz problem 95

4 alpha-interpolation of fem and fdm 97

4.1 Introduction 97

4.2 Problem statement 101

4.3 Analysis in 1d 101


4.3.2 α-Interpolation of the Galerkin-FEM and the classical FDM 102

4.3.3 Dispersion plots in 1d 104

4.3.4 Examples 104



4.4.2 Galerkin FEM using rectangular bilinear finite elements 108

4.4.3 A nonstandard compact stencil in 2d 109

4.4.4 Numerical solution, phase error and local truncation error 112

4.4.5 α-Interpolation of the FEM and the FDM in 2d 115

4.4.6 Dispersion plots in 2d 116

4.4.7 Examples 118

4.5 Conclusions and Outlook 124

5 petrov–galerkin formulation 129

5.1 Introduction 129

5.2 Mass lumping 130

5.3 Variational setting 134

5.4 Block finite elements 138

5.4.1 1d linear FE 138

5.4.2 2d bilinear FE 140

5.4.3 Stabilization Parameters 141

5.5 Simplicial finite elements 142

5.6 Examples 144

5.6.1 Example 1: Dirichlet boundary conditions 146

5.6.2 Example 2: Robin boundary conditions 147

5.7 Conclusions 150

iii stokes problem 155

6 pressure laplacian stabilization 157

6.1 Introduction 157

6.2 Governing equations 159

6.3 Integral form of the momentum equations 160

contents xiii

6.4 Stabilized form of the incompressibility equation using finite calcu-lus 160

6.4.1 Finite calculus 161

6.4.2 First order FIC form of the incompressibility equation 163

6.4.3 Higher order FIC form of the incompressibility equation 163

6.5 On the proportionality between the pressure and the volumetric strainrate 164

6.6 A penalty-type stabilized formulation 164

6.7 Galerkin-least squares (GLS) formulation 165

6.8 Pressure Laplacian stabilization (PLS) method 166

6.8.1 Variational form of the mass balance equation in the PLS method 166

6.8.2 Computation of the stabilization parameters in the PLS method 167

6.8.3 PLS boundary stabilization term 168

6.9 Pressure-gradient projection (PGP) formulation 169

6.10 Orthogonal sub-scales (OSS) formulation 170

6.11 PLS+π method 171

6.12 Finite element discretization 171

6.12.1 Discretized equations 171

6.12.2 Solution schemes for penalty, GLS, PLS methods 174

6.12.3 Solution scheme for PGP and OSS methods 174

6.13 Definition of the characteristic lengths 175

6.14 Some comments on the different stabilization methods 176

6.14.1 Penalty method 176

6.14.2 GLS method 176

6.14.3 PLS method 176

6.14.4 PGP and OSS method 176

6.14.5 Comparison between the different stabilization methods 176

6.15 Examples of application 177

6.15.1 Hydrostatic flow problem for a single fluid in a square domain 178

6.15.2 Two-fluid hydrostatic problem in a square domain 179

6.15.3 Poiseuille flow in a trapezoidal domain 180

6.15.4 Lid driven cavity problem 180

6.15.5 Manufactured flow problem in a trapezoidal domain 181

6.16 Conclusions 184

conclusions 189

bibliography 193

L I S T O F F I G U R E S

Figure 1 Plot of ω∗h vs ξ∗ for the semi-discrete problem. 16

Figure 2 Amplification plots of the semi-discrete problem for the FIC/-SUPG and FIC_RC/OSS methods. 17

Figure 3 Contour plot of log10(<(|ω∗h − ω∗|)) for the backward Eulerscheme. 18

Figure 4 Contour plot of log10(=(ω∗h)) for the backward Euler scheme. 19

Figure 5 Contour plot of log10(<(|ω∗h −ω∗|)) for the Crank-Nicholsonscheme. 20

Figure 6 Contour plot of log10(=(ω∗h)) for the Crank-Nicholson scheme. 21

Figure 7 Contour plot of log10(<(|ω∗h−ω∗|)) for the BDF2 scheme. 22

Figure 8 Contour plot of log10(=(ω∗h)) for the BDF2 scheme. 23

Figure 9 Real part of the normalized group velocity ∂ω∗h/∂ξ∗ vs. ξ∗ for

the Crank Nicholson scheme. 24

Figure 10 Plot of optimal α (for the interior nodes) vs. γ for the FIC_RC/OSSmethod. 26

Figure 11 Node stencil for the 1d problem. 26

Figure 12 Plot of optimal α (for the penultimate boundary nodes) vs. γ forthe FIC_RC/OSS method. 27

Figure 13 Plot of optimal α (for the elements adjacent to the boundary) vs.γ for the FIC_RC/OSS method. 28

Figure 14 Example 1: transport of a Gaussian pulse. 31

Figure 15 Example 2: transport of a Gaussian and square pulse. 32

Figure 16 Example 3: transport of a sinusoidal wave. 34

Figure 17 Example 4: steady-state solutions on uniform and nonuniformgrids for the FIC/SUPG and FIC_RC/OSS methods. 35

Figure 18 The design problem and monotone solution. 45

Figure 19 The plot of g(η) for η ∈ [0, 1]. 47

Figure 20 Example 1: HRPG Type I, η1 = 0.5,η2 = 0.3. 49

Figure 21 Example 1: HRPG Type II, η1 = 0.5,η2 = 0.3. 49

Figure 22 Example 1: HRPG Type I, η1 = 1.0,η2 = 0.0. 50

Figure 23 Example 1: HRPG Type II, η1 = 1.0,η2 = 0.0. 50

Figure 24 Example 2: HRPG Type I. 51

Figure 25 Example 2: HRPG Type II. 52

Figure 26 Example 3: HRPG Type I, total variation plots. 52

Figure 27 Example 3: HRPG Type II, total variation plots. 53

Figure 28 Steady state: (γ,ω, f,φpL,φpR) = (1, 5, 0, 8, 3). 61







xiv

List of Figures xv



Figure 37 Steady state: (γ,ω, f,φpL,φpR) = (2, 0,u, 0, 0). 65

Figure 38 Transient case: pure convection example set-1 67

Figure 39 Transient case: pure convection example set-2 68

Figure 40 Solution of a singularly perturbed convection–diffusion prob-lem. 71

Figure 41 Parabolic layers in the solution of the heat equation and thediffusion–reaction problem. 73

Figure 42 Matching the parabolic layers in the solution of the heat equa-tion and the diffusion–reaction problem. 73

Figure 43 Anisotropic element length vectors for the 2d bilinear block fi-nite element. 75

Figure 44 Unstructured 20× 20 meshes made of bilinear block finite ele-ments. 79

Figure 45 Example 1, advection skew to the mesh. 80

Figure 46 Example 2, nonuniform rotational advection. 81

Figure 47 Example 3, a reaction–diffusion problem. 82

Figure 48 Example 4, a convection–diffusion–reaction problem. 83

Figure 49 Example 5, uniform advection with a constant source term. 84

Figure 50 Example 6, non-uniform advection with a constant source term. 85

Figure 51 Example 7, uniform advection with a discontinuous source term. 86

Figure 52 Initial data for the transient 2d advection examples. 87

Figure 53 Example 8, transient pure convection skew to the mesh, eleva-tion plots. 88

Figure 54 Example 8, transient pure convection skew to the mesh, contourplots. 89

Figure 55 Example 9, rotation of solid bodies, elevation plots. 90

Figure 56 Example 9, rotation of solid bodies, contour plots. 91

Figure 57 Example 10, uniform advection with a negative source term 92

Figure 58 Plots of f(ω∗) and ξ∗(ω∗). 105

Figure 59 A schematic diagram of a zone of degeneracy. 106

Figure 60 Numerical solutions obtained using at least 8, 16, 32 and 64 ele-ments per wavelength. 107

Figure 61 A schematic diagram of the contours traced by the numericalsolution Ph(ξh1 `, ξ

h2 `) and the exact solution P(ξβ1 `, ξ

β2 `). 113

Figure 62 ξ∗1 − ξ∗2 contours for ω∗ ∈ (1/4), (1/9), (1/16), (1/25) and ω∗1 =

ω∗2. 119

Figure 63 ξ∗1 − ξ∗2 contours for ω∗ ∈ (1/4), (1/9), (1/16), (1/25) and ω∗2 =

0.49ω∗1. 120

Figure 64 Relative local truncation error plots. 121

Figure 65 Log-scaled relative local truncation error plots. 122

Figure 66 Convergence of the relative error for ξo = 10√10. 125

Figure 67 Convergence of the relative error for ξo = 100. 126

Figure 68 Convergence of the relative error in the l∞ Euclidean norm. 127

Figure 69 Test functions corresponding to the 1d linear FE that results inmass lumping. 133

Figure 70 Comparison of the test function w and trial solution φ of ageneric Discontinuous–Galerkin method with those of the cur-rent Petrov–Galerkin method. 134

Figure 71 Schematic diagrams of an arbitrary element K ∈ Th and a testfunction defined over it. 135

Figure 72 A model for the PG weights on the element edges correspondingto the 1d linear FE. 139

Figure 73 Stencils obtained by using a structured simplicial finite elementmesh. 142

Figure 74 Meshes made of bilinear block finite elements. 145

Figure 75 Convergence of the relative error in the L2 norm using β = (π/9)

and Dirichlet boundary conditions. 148

Figure 76 Convergence of the relative error in the H1 semi-norm usingβ = (π/9) and Dirichlet boundary conditions. 149

Figure 77 Convergence of the relative error in the l∞ Euclidean norm us-ing β = (π/9) and Dirichlet boundary conditions. 150

Figure 78 Convergence of the relative error in the L2 norm using β = (π/9)

and Robin boundary conditions. 151

Figure 79 Convergence of the relative error in the H1 semi-norm usingβ = (π/9) and Robin boundary conditions. 152

Figure 80 Convergence of the relative error in the l∞ Euclidean norm us-ing β = (π/9) and Robin boundary conditions. 153

Figure 81 Finite-size balance domain used in FIC. 161

Figure 82 Solution of the hydrostatic flow problem in a square domain. 179

Figure 83 Convergence of the PLS method for the two-fluid hydrostaticproblem. 179

Figure 84 Trapezoidal domain discretized by a symmetrical mesh of 2×10× 10 3-node triangles. 180

Figure 85 Poiseuille flow in a trapezoidal domain: pressure plots. 181

Figure 86 Poiseuille flow in a trapezoidal domain: velocity plots. 182

Figure 87 Convergence of the PLS solution for the Poiseuille flow prob-lem. 183

Figure 88 Lid driven cavity problem: pressure elevation plots. 185

Figure 89 Lid driven cavity problem: pressure contour plots. 186

Figure 90 Convergence of PLS results for the lid driven cavity problem. 187

Figure 91 Manufactured flow problem: convergence rates for the GLS, OSS,PLS and PLS+π methods. 187

L I S T O F TA B L E S

Table 1 Matrix definitions for FIC and FIC_RC methods 12

xvi

List of Tables xvii

Table 2 Perturbations associated with Petrov–Galerkin methods 41

Table 3 Test functions corresponding to some finite elements 132

Let auspicious thoughts come unto us from every direction.

— Rig Veda I:89.1.

I N T R O D U C T I O N

This monograph consists of three parts. The first part describes the work done onthe convection–diffusion–reaction problem, the second one deals with the Helmholtzproblem and the third one deals with the Stokes problem. In each part we proposea new stabilized finite element based Petrov–Galerkin method for the correspondingproblem. The work presented in each part is independent from the rest of the partsand hence can be read arbitrarily.

The motivation for the current work is in the quest to develop new numerical meth-ods capable of providing stable and accurate solutions to the problems of fluid me-chanics and their interaction with structures. The work presented here is the firststep towards this objective. This is done by the study of the convection–diffusion–reaction, the Helmholtz and the Stokes problems. These problems are of vital impor-tance as they are the simplest models related to transport processes, wave propagationphenomenon and associated numerical difficulties that arise in fluid flow problemsand fluid-structure interaction. The convection–diffusion–reaction equation is an idealmodel problem to study the stabilization of singularly perturbed problems. Ten typi-cal problems where the convection–diffusion phenomenon occurs is listed in [134]. Forinstance, a common source is the linearization of Navier–Stokes equations with largeReynolds number. The Helmholtz equation is an ideal model problem to study theso-called pollution effect. It is also the simplest model problem concerning wave prop-agation phenomenon, viz. acoustics, elastodynamics, fluid-structure interaction, elec-trodynamics etc. [106]. A simple one-dimensional fluid-structure interaction problemmodeled using the Helmholtz equation was presented in [47, Section 5]. The Stokesproblem is the simplest model describing incompressible flow of a viscous fluid andthus are ideal to study pressure stabilization, i. e.circumventing the div-stability condi-tion (also known as the Ladyzhenskaya-Babuska-Brezzi condition) that the finite elementspaces should otherwise satisfy. We refer to [75, Chapter 2] for an exposition of thesame.

The point of departure was initially set to the analysis of an approach that hasbeen gaining momentum in the literature: consistency recovery procedures1 for lower-order stabilized finite element methods. For residual-based stabilization methods, thehigher-order derivatives of the residual that appear in the stabilization terms vanishwhen lower-order finite elements are used. Consistency recovery methods have beenadvocated for these cases and have been shown to result in improved accuracy forsome problems [111, 148]. To be precise, the improved accuracy was reported for otherrelated unknowns of the problem (e. g.the pressure field) and not for the transportedunknown (e. g.the velocity field). The first chapter of this monograph discusses thegain/loss by recovering the consistency of the discrete residual in the stabilizationterms via the form that includes the convective projection variable (as in the OSS

1 also know as residual correction methods

1

2 introduction

method [34]) for the 1d convection-diffusion equation. The dispersion analysis of thesemi-discrete and fully-discrete problems was done and no gain in the dispersionaccuracy is found by including the convective projection variable. Further, the optimalexpression of the stabilization parameter on uniform meshes for the steady-state caseand including the convective projection variable revealed a strategical difficulty inthe development of discontinuity capturing methods. Hence, we do not prefer thisconsistency recovery method in the stabilization of singularly perturbed problems.

In the second chapter we present the design of a high-resolution Petrov–Galerkin(HRPG) method for the 1d convection–diffusion–reaction problem. The problem isstudied from a fresh point of view, including practical implications on the formula-tion of the maximum principle, M-Matrices theory, monotonicity and total variationdiminishing (TVD) finite volume schemes. The prefix high-resolution is used herein the sense popularized by Harten [82] in the finite-difference and finite-volumecommunity, i. e.second order accuracy for smooth/regular regimes and good shock-capturing in non-regular regimes. The HRPG method is designed using the divideand conquer strategy, i. e.the original problem is further divided into smaller modelproblems where the different types of numerical artifacts that plague the originalproblem are singled-out and the expressions for the stabilization parameters are de-rived/updated to treat them effectively. Several 1d examples are presented that sup-port the design objective—stabilization with high-resolution.

In the third chapter we present a multi dimensional extension of the HRPG methodusing multi-linear block finite elements. In higher dimensions the solutions to theconvection–diffusion–reaction problem might additionally develop characteristic lay-ers. These layers are a unique feature of multi dimensions and hence have no in-stances in 1d. So we design a nondimensional element number that quantifies thecharacteristic layers. By quantification we mean that it should serve a similar purposein the definition of the stabilization parameters as the element Peclet number doesfor the exponential layers. Although the structure of HRPG method in 1d is identicalto the consistent approximate upwind Petrov–Galerkin method [68], in multi dimen-sions the former method has a unique structure. The distinction is that in general theupwinding is not streamline and the discontinuity-capturing is neither isotropic norpurely crosswind. In this line, we present anisotropic element length vectors and usingthem objective characteristic tensors associated with the HRPG method are defined.Except for the modification to include the new dimensionless number that quanti-fies the characteristic layers, the definition of the stabilization parameters calculatedalong the element length vectors are a direct extension of their counterparts in 1d. Thestrategy used to treat the artifacts about the characteristic layers is to treat them justlike the artifacts found across the parabolic layers in the reaction-dominant case. Sev-eral 2d examples are presented that illustrate not only the advantages of the HRPGmethod but also its limitations. Of course, the advantages outscore the limitations.

The second part of this monograph deals with the work on the Helmholtz problem.Although this equation is related to some fluid-structure interaction problems, theevents that lead to the work presented here had a modest kickoff. Chronologically, thework began just after the development of the HRPG method for the 1d convection–diffusion–reaction problem. As the properties of the Helmholtz equation is sharedby the diffusion–production problem (obtained using a negative reaction co-efficient),we were curious to investigate if the HRPG form be efficient to solve this case. As

introduction 3

the HRPG form is nonlinear, one needs to choose a target solution to design thestabilization parameters so as to recover this target solution. In 1d the simplest targetsolution is the one that you get with the Galerkin FEM using a higher-order massmatrix [70, 71, 110]. This scheme has a dispersion accuracy of fourth-order in 1d butin higher dimensions it drops to second-order. Unfortunately all attempts to recoverthis target solution within the HRPG form were in vain. Although this exercise boreno fruit in favor of the HRPG method, we discovered alternate target solutions for theHelmholtz equation that had higher-order dispersion accuracy in multi dimensions.

In the fourth chapter we present some observations and related dispersion anal-ysis of a simple domain-based higher-order compact numerical scheme (involvingtwo parameters) for the Helmholtz equation. The stencil obtained by choosing the pa-rameters as distinct was denoted therein as the ’nonstandard compact stencil’. Mak-ing the parameters equal, the nonstandard compact stencil simplifies to the alpha-interpolation of the equation stencils obtained by the Galerkin finite element method(FEM) and classical central finite difference method (FDM). For the Helmholtz equa-tion, generic expressions for the parameters were given that guarantees a dispersionaccuracy of sixth-order should the parameters be distinct and fourth-order shouldthey be equal. Convergence studies of the relative error in the L2 norm, the H1 semi-norm and the l∞ Euclidean norm are done and the pollution effect is found to besmall.

In the fifth chapter we present a new Petrov–Galerkin method involving two param-eters that yields on rectangular meshes the nonstandard compact stencil presented ear-lier in the fifth chapter. This Petrov–Galerkin method provides the counterparts of thelater scheme on unstructured meshes and allows the treatment of natural boundaryconditions (Neumann or Robin) and the source terms in a straight-forward manner.First, we present the test-functions that reproduce the mass matrix lumping techniquewithin a Petrov–Galerkin setting. Then, we show that the use of these test functionsin an appropriate variational setting reproduces the FDM stencil for the current prob-lem on rectangular meshes. Next, we show that an appropriate combination of thesetest-functions with the standard FEM shape functions will yield the nonstandard com-pact stencil on rectangular meshes. Convergence studies of the relative error in theL2 norm, the H1 semi-norm and the l∞ Euclidean norm show that the proposedPetrov–Galerkin method inherits the higher-order dispersion accuracy observed forthe aforesaid numerical scheme.

The third and final part of this monograph deals with the work done on the Stokesproblem. Here we present a collection of stabilized finite element (FE) methods de-rived via first-order and second-order finite calculus (FIC) procedures. It is shown thatseveral well known existing stabilized FE methods such as the penalty technique, theGalerkin Least Square (GLS) method, the Pressure Gradient Projection (PGP) methodand the orthogonal sub-scales (OSS) method are recovered from the general residual-based FIC stabilized form. A new family of Pressure Laplacian Stabilization (PLS)FE methods with consistent nonlinear forms of the stabilization parameters are de-rived. The distinct feature of the family of PLS methods is that they are nonlinear andresidual-based, i. e.the stabilization terms depend on the discrete residuals of the mo-mentum and/or the incompressibility equations. The advantages and disadvantagesof these stabilization techniques are discussed and several examples of application arepresented.

Part I

C O N V E C T I O N - D I F F U S I O N - R E A C T I O N P R O B L E M

Experience is what you get when you don’t get what you want.

— Dan Stanford.

1A N A LY S I S O F A C O N S I S T E N C Y R E C O V E RY M E T H O D

1.1 introduction

In many transport processes arising in physical problems, convection essentially dom-inates diffusion. The design of numerical methods for such problems that reflect theiralmost hyperbolic nature and guarantee that the discrete solution satisfies the physicalconditions is a subject that has been widely studied. In particular for the convection-diffusion problem the standard Galerkin finite element method leads to numericalinstabilities for the convection dominated case. Several stabilization methods, for in-stance the Streamline-Upwind Petrov-Galerkin (SUPG), Galerkin Least Square (GLS),Sub-Grid Scale (SGS), SGS with Orthogonal Sub-scales (OSS) etc., have been designedto overcome this numerical instability. A thorough comparison of some of these meth-ods from the point of view of their formulation and the motivations that lead to themcan be found in [32]. Also stabilization procedures based on Finite Calculus (FIC) havebeen developed as a general purpose tool for improving the stability and accuracy ofthe convection-diffusion problem [141, 143, 145, 147]. A residual correction methodbased on FIC was presented in [148] and is shown to yield an equivalent formulationto an OSS form [34] with very little manipulation.

For the convection diffusion problem using the SUPG or FIC methods, the higherorder term (here the diffusion term) that appears in the stabilization term vanishwhen simplicial elements are used. In [148] it is shown that for the elasticity prob-lem, the form that includes the projected gradient of pressure into the stabilizationterms (motivated by the OSS method) is essential to obtain accurate numerical resultswhich converge in a more monotone manner and are less sensitive to the value ofthe stabilization parameter. In [111] a method was presented to globally reconstructa continuous approximation to the diffusive flux for linear finite elements using a L2

projection and shown, in some cases (when advection and diffusion are on a par), togreatly improve accuracy. It is important to note that again this improved accuracy isdemonstrated for other related unknowns of the problem (like pressure) and not forthe transported unknown (say velocity). Also when convection dominates diffusionthere is little effect in the inclusion of the recovered diffusive flux. On the other hand,consistency recovery following the OSS philosophy is independent from the diffusiveterm. These observations are the motivation to investigate in detail the benefits ofincluding similar projections for the convection diffusion problem following the OSSphilosophy. In other words, we try to answer the question - what do we gain/looseby recovering the consistency of the discrete residual in the stabilization terms for theconvection-dominated case via the introduction of OSS-type convective projection ?

The von Neumann analysis for the Galerkin and SUPG method is relatively wellknown [72]. Relevant literature on the type of analysis presented here may be found

7

8 1 analysis of a consistency recovery method

in [26]. First, we present the von Neumann analysis for the 1d FIC method with re-covered consistency (FIC_RC). This is achieved by including the convective projectioninto the stabilization term (motivated by the OSS method). It is then shown that in1d the FIC and FIC_RC methods are equivalent to the SUPG and OSS methods re-spectively. Consequently the comparisons made between the former methods may becarried over to the later methods. The transient analysis is done by examining the dis-crete dispersion relation (DDR) of the stabilization methods. The explanation for theoccurrence of wiggles/oscillations in the transient evolution of the numerical solutionwas explained by examining the dispersion relations of the continuous and discreteproblems in [183]. It has been found that beyond a certain wavenumber ξd the con-tinuous and the discrete dispersion relations diverge [44, 183]. This wavenumber (ξd)is referred to as the phase departure wave number in [44]. If the bandwidth of the ampli-tude spectra of any given initial function has wavenumbers greater than ξd, the initialfunction suffers a change of form (with wiggles/oscillations) in its transient evolution.Examining the respective DDRs, we seek to find if the stabilization methods provideany improvements in the solutions. A comparison of the DDR of the FIC_RC/OSSmethod is done with the DDRs of the Galerkin and FIC/SUPG methods for three stan-dard time integration schemes. Also, the range of wavenumbers to which the DDRagrees with the continuous dispersion relation is shown to extend, should a consistent“effective” mass matrix be preferred to a lumped one. Next, it is shown that unlikethe FIC/SUPG method, the FIC_RC/OSS method introduces a certain rearrangementin the equation stencils at nodes on and adjacent to the domain boundary. Thus usinga uniform expression for the stabilization parameter (α) will lead to enhanced local-ized oscillations at the boundary. For the 1d steady-state problem , we present a newexpression for α which is optimal for uniform grids and provides negligible dampingwhen used in the transient mode. Unfortunately for non-uniform grids, it leads toweak node-to-node oscillations.

1.2 transient convection diffusion equation

1.2.1 Problem Statement

The statement of the multi-dimensional problem is as follows:

∂φ

∂t+ u ·∇φ−∇ · (k∇φ) − f = 0 in Ω (1.1a)

φ(x, t = 0) = φo(x) in Ω (1.1b)

φ = φp on ΓD (1.1c)

k∇φ · n = qp on ΓN (1.1d)

where, u is the convection velocity, k is the diffusion, f is the source, φ(x, t) is thetransported variable, φo(x) is the initial solution, φp and qp are the prescribed valuesof φ and the diffusive flux at the Dirichlet and Neumann boundaries respectively. The

1.2 Transient Convection Diffusion Equation 9

FIC formulation of this problem (neglecting the time stabilization terms [145]) is asfollows:

r−1

2h ·∇r = 0 in Ω (1.2a)

φ(x, t = 0) = φo(x) in Ω (1.2b)

φh = φp on ΓD (1.2c)

k∇φh · n +1

2(h · n)r = qp on ΓN (1.2d)

r :=∂φ

∂t+ u ·∇φ−∇ · (k∇φ) − f (1.2e)

Let(·, ·)

and(·, ·)ΓN

denote the L2(Ω) and L2(ΓN) inner products respectively. Thevariational form of the problem (1.1) can be expressed as follows: Find φ : [0, T ] 7→ V

such that ∀w ∈ V0 we have,

a(w,φ

)= l(w)

(1.3a)

a(w,φ

):=(w,∂φ

∂t

)+(w, u ·∇φ

)+(∇w,k∇φ

)(1.3b)

l(w):=(w, f

)+(w,qp

)ΓN

(1.3c)

where V := w : w ∈ H1(Ω) and w = φp on ΓD, V0 := w : w ∈ H1(Ω) and w =

0 on ΓD. For the FIC equations the variational form can be expressed as follows: Findφ : [0, T ] 7→ V such that ∀w ∈ V0 we have,

a(w,φ

)+1

2

(∇ · (hw), r

)= l(w)

(1.4)

The calculation of the residual contribution in the stabilization term can be simpli-fied if we introduce the projection of the convective term π via an auxiliary equationdefined as,

π = u ·∇φ− r (1.5)

We express the residual, r, that occurs in the stabilization term(∇ · (hw), r

)as a

function of π. Thus π becomes an addendum to the set of unknowns to be found.The integral equation system is now augmented forcing that the residual r expressedin terms of π via Eq.(1.5) vanishes (in average) over the analysis domain. Problem(1.4) after the addendum of the convective projection π is expressed as follows: Findφ : [0, T ] 7→ V and π ∈ H1(Ω) such that ∀(w, z) ∈ (V0(Ω),H1(Ω)),

a(w,φ

)+1

2

(∇ · (hw), u ·∇φ− π

)= l(w)

(1.6a)(z,π)=(z, u ·∇φ

)(1.6b)

We remark that the projection of the convective term provides consistency to theformulation, i. e.the system of equations (1.6) have the residual form which vanishesfor the exact solution. Henceforth we refer to the formulation given by the equation(1.6) as the FIC formulation with recovered consistency (FIC_RC). The convectiveprojection is expected to capture the otherwise lost effect of higher order terms inthe residual when simplicial elements are used. The introduction of the convectiveprojection variable π was deduced from the OSS approach in [34].


1.2.2 Dispersion Relation in 1d

Any equation that admits plane wave solutions of the form exp[i(ωt− ξx)], but withthe property that the speed of propagation of these waves is dependent on ξ, is gen-erally referred to as a dispersive equation. Here ξ, ω are the angular wave-number andthe frequency, respectively. The equation that expresses ω as a function of ξ is knownas the dispersion relation. Generally the transient convection-diffusion equation is a dis-persive equation. In the limit, when diffusion tends to zero and the equation morphsinto a pure convection problem, it tend to become non-dispersive [183]. In 1d and forthe sourceless case (f = 0), Eq.(1.1a) can be expressed as follows:

∂φ

∂t+ u

∂φ

∂x− k

∂2φ

∂x2= 0 (1.7)

For a given discretization of size ` in space and an increment θ in time, we expressthe Courant and Peclet numbers as C = uθ

` and γ = u`2k . We write down the dispersion

relation (Eq. 1.8) for the continuous problem (Eq. 1.7) by propagating the plane wavesolution φ =exp[i(ωt − ξx)]. From Eq.(1.8) we obtain, taking the limit k → 0 orγ→∞, the dispersion relation for the pure convection problem.

ω = uξ+ ikξ2 (1.8a)

ωθ = C(ξ`) + iC

2γ(ξ`)2 (1.8b)

Now let us consider the amplification of the solution from time step n to n+ 1 andat some given spatial point. The amplification parameter β as defined in [44] is given byEq.(1.9). It can be clearly seen that β is stationary in time and uniform in space. Theamplification and phase shift per time step are given by the magnitude and argumentof Eq.(1.9), respectively. Thus for the pure convection problem we can see that theamplification is unity, i. e.|β| = 1. The phase and group velocities are given by the Eqs.(1.10a) and (1.10b) respectively [73].

β =φn+1j

φnj= exp [iωθ] = exp

[−kθξ2

]exp [iuθξ] = exp

[−C

2γ(ξ`)2

]exp [iC(ξ`)] (1.9)

Vp(ξ) =ω(ξ)

ξ(1.10a)

Vg(ξ) =∂ω(ξ)

∂ξ(1.10b)

1.3 fe discretization

1.3.1 Semi-Discrete Form

The semi-discrete (continuous in time, discrete in space) counterpart of the FIC method(1.4) can be written as follows: Find φh : [0, T ] 7→ Vh such that ∀wh ∈ Vh0 we have,

a(wh,φh

)+∑e

1

2

(∇ · (hwh), rh

)Ωe

= l(wh)

(1.11)


where Vh ⊂ V and Vh0 ⊂ V0. The stabilization term in Eq.(1.11) has been expressedas a sum of the element contributions to allow for inter-element discontinuities inthe term ∇rh of Eq.(1.2), where rh := r(φh) is the residual of the FE approximationof the infinitesimal governing equation and

(·, ·)Ωe

denote the L2(Ωe) inner product.Similarly the discrete counterpart of the FIC_RC method (1.6) can be written as: Findφh : [0, T ] 7→ Vh and πh ∈ H1(Ω) such that ∀(wh, zh) ∈ (Vh0 (Ω),H1(Ω)),

a(wh,φh

)+∑e

1

2

(∇ · (hwh), u ·∇φh − πh

)Ωe

= l(wh)

(1.12a)

(zh,πh

)=(zh, u ·∇φh

)(1.12b)

The variables in the Eqs.(1.11) and (1.12) interpolated by finite element shape func-tions Na can be expressed as follows:

φh = Naφa,πh = Naπa,wh = Nawa, zh = Naza (1.13)

where a is the spatial node index. The discrete problems (1.11) and (1.12) can bewritten in matrix notation via Eqs.(1.14) and (1.15), respectively, as follows:

[M + S2]Φ+ [C + D + S1+ S3]Φ = fg + fs (1.14)

MΦ+ [C + D + S1]Φ− S2Π = fg (1.15a)

MΠ− CΦ = 0 (1.15b)

where Φ := φa and Π := πa represent the vector of nodal unknowns. Theelement contributions to the matrices and vectors in Eqs.(1.14) and (1.15) are given by

Ceab =(Na, u ·∇Nb

)Ωe

, Deab =(∇Na,k∇Nb

)Ωe

(1.16a)

Meab =

(Na,Nb

)Ωe

, S1eab =1

2

(∇ · (hNa), u ·∇Nb

)Ωe

(1.16b)

S2eab =1

2

(∇ · (hNa),Nb

)Ωe

, S3eab = −1

2

(∇ · (hNa),∇ · (k∇Nb)

)Ωe

(1.16c)

fgea =(Na, f

)Ωe

+(Na,qp

)ΓN

, fsea =1

2

(∇ · (hNa), f

)Ωe

(1.16d)

Note that Eq.(1.15b) correspond to the L2-projection of the term u ·∇φh onto thespace spanned by the shape functions. Whenever the later term admits discontinuitiesthe projection is non-monotone [34]. A monotone projection of the convective termcan be achieved if the mass matrix M that appears in Eq.(1.15b) is lumped. Also theexpression for Π will have local support only when M is lumped. This feature allowsus to study generic nodal equation stencils for the interior of the domain. Henceforthwe always consider the Eq.(1.15b) with M lumped. Eqs.(1.14) and (1.15) may be ex-pressed in a general form as shown in Eq.(1.17). Table 1 defines the correspondingmatrices for the FIC and FIC_RC methods.

TΦ+ [C + D + S]Φ = f (1.17)


FIC FIC_RC

T-lumped ML + S2 ML

T-consistent M + S2 M

S S1+ S3 S1− S2M−1L C

f fg+fs fg

Table 1: Matrix definitions for FIC and FIC_RC methods

where matrix T is the ’effective’ mass matrix.Note that for the direction of h being the same as that of the velocity u, i. e.h =

2τu and assuming τ constant within an element, the form of the stabilization term(∇ · (hwh), rh

)Ωe

in Eq.(1.11) is identical to that of the standard SUPG method. Thuswith this choice of h the FIC and FIC_RC methods are identical to the SUPG andOSS (orthogonal sub-scales) methods respectively. The general direction of h intro-duces naturally stabilization along the streamlines and also along the directions ofthe gradient of the solution transverse to the velocity vector. The FIC formulationtherefore incorporates the best features of the SUPG and the shock-capturing meth-ods. Applications of the FIC-FEM formulation to a wide range of convection-diffusionproblems with sharp gradients are presented in [154]. We remark that in 1d (assum-ing h constant within an element) the FIC and the FIC_RC methods are identical tothe SUPG and OSS methods respectively. Thus the conclusions made between the FICand FIC_RC methods may be carried over to those between SUPG and OSS methods.

1.3.2 DDR in 1d

The DDR of the semi-discrete problem (semi-DDR) and when the temporal termsare discretized using two class of time discretization schemes are investigated in thissection. The time discretization schemes considered are the trapezoidal scheme and thesecond-order backward differencing formula (BDF2). The effects on the numerical disper-sion due to the choice of the form of the effective mass matrix T (lumped or consistent)in Eq.(1.17) are also studied. The flag lumped or consistent refers only to the matrix Tas defined in Table 1 for the FIC and FIC_RC methods. The DDRs are written by in-serting a plane wave solution of the form φ = exp[i(ωt− ξx)] into the correspondingequation stencils. Taking the limit γ→∞ we recover the DDR for the pure convectionproblem.

1.3.2.1 Semi-DDR

We study the equation stencil for an interior node of the semi-discrete problem givenby Eq.(1.17) with f = 0 in 1d. For a compact representation of the stencils we introducethe following definition,

(?) := (u

2)(φj+1 −φj−1) − (

k

`+uα

2)(φj+1 − 2φj +φj−1) (1.18)


FIC/SUPG method:

(`

6)[(−3α

2)φj+1 + 6φj + (

3α

2)φj−1

]+ (?) = 0, T-lumped (1.19a)

(`

6)[(1−

3α

2)φj+1 + 4φj + (1+

3α

2)φj−1

]+ (?) = 0, T-consistent (1.19b)

FIC_RC/OSS method:

` φj + (?) + (uα

8)(φj+2 − 2φj +φj−2) = 0, T-lumped

(1.20a)

(`

6)(φj+1 + 4φj + φj−1) + (?) + (

uα

8)(φj+2 − 2φj +φj−2) = 0, T-consistent

(1.20b)

Making α = 0 in Eqs. (1.19) and (1.20) we recover the standard Galerkin method.The Semi-DDRs for all the methods and for the T-lumped, T-consistent cases can beexpressed in a generic and compact manner as follows:

ωh = −iB

θA(1.21)

where,

A :=

1

CGalerkin, FIC_RC/OSS methods, T-lumped

2+ cos(ξ`)3C

Galerkin, FIC_RC/OSS methods, T-consistent

1

C+ i

α

2Csin(ξ`) FIC/SUPG methods, T-lumped

2+ cos(ξ`)3C

+ iα

2Csin(ξ`) FIC/SUPG methods, T-consistent

(1.22a)

B :=

i sin(ξ`) − 2 sin2(ξ`

2)

(1

γ

)Galerkin method


2)

(1

γ+α

)FIC/SUPG methods


2)

(1

γ+α sin2(

ξ`

2)

)FIC_RC/OSS methods

(1.22b)

1.3.2.2 Trapezoidal Scheme

The structure of the equation stencil for an interior node with respect to the spatialindices is the same as in the semi-discrete problem. Henceforth we express the fullydiscrete system in the matrix notation only.

T · Φn+1 −Φn

θ+ [C + D + S] ·Φn+σ = 0 (1.23a)

Φn+σ := σ Φn+1 + (1− σ) Φn (1.23b)


Making σ = 0, 0.5, 1 we recover the forward Euler, Crank-Nicholson and backward Eulerschemes respectively. The DDR for the trapezoidal scheme can be expressed in termsof A and B defined in Eq.(1.22) as follows,

exp [iωhθ] =A+ (1− σ)B

A− σBor equivalently, (1.24a)

tan(ωhθ

2

)= −i

B

2A+ (1− 2σ)B(1.24b)

1.3.2.3 BDF2 Scheme

The fully discrete system of equations after time discretization by the BDF2 scheme isgiven by,

T · 3Φn+1 − 4Φn +Φn−1

2θ+ [C + D + S] ·Φn+1 = O (1.25)

The DDR for the BDF2 scheme is a quadratic relation in exp[iωhθ]. The solution tothe quadratic equation gives two expressions for the DDR, which can be expressed asfollows:

exp [iωhθ] =2A+

√A2 + 2AB

3A− 2B(1.26a)

exp [iωhθ] =2A−

√A2 + 2AB

3A− 2B(1.26b)

We remark that the solution given by Eq.(1.26b) predicts negative values of <(ωh)1

for positive wave-numbers. Thus, we consider the solution given by Eq.(1.26a) as theonly acceptable solution.

1.3.3 DDR Plots

The DDRs presented in the previous section represent the frequency as a functionof six independent variables, i. e.ωh := ωh(ξ, `, θ,C,γ,α). For a feasible graphicalrepresentation of the DDRs we freeze some of them and normalize the frequencyand wavenumbers to retain maximum generality. In the DDR plots we consider onlythe pure convection problem (k = 0). The stabilization parameter α = 1.0 is chosen.This corresponds to the optimal value for the SUPG/FIC method in 1d and for auniform mesh. This choice is made for convenience and comparison of the effectsof the stabilization term introduced by the considered methods. Note that the DDRsare periodic in ξ and the corresponding fundamental domain is ξ ∈ [−π/`,π/`]. TheNyquist frequency in space is ξnq = π/` and in time is ωnq = π/θ. Thus in the DDRplots we do not consider wavenumbers and frequencies beyond the Nyquist limits. Itcan be shown with respect to the exact dispersion relation (Eq. 1.8) that this conditioncorresponds to choosing C 6 1. We normalize the wavenumber ξ by the Nyquistlimit ξnq, i. e.ξ∗ = ξ/ξnq. The frequency ωh is normalized as ω∗h = ωh/(uξnq) =

ωhθ/(Cπ) = ωh`/(uπ). The DDRs are now expressed with respect to the normalizedwavenumber and frequency, i. e.ω∗h := ω∗h(ξ

∗,C). The plotting domain considered is(ξ∗,C) = (0, 1)× (0, 1).

1 The real and imaginary parts of ωh are denoted as <(ωh) and =(ωh), respectively


1.3.3.1 Semi-discrete case

For the semi-discrete problem the DDR no longer depends on θ. The frequency ω∗his now only a function of ξ∗, i. e.ω∗h := ω∗h(ξ

∗). The amplification at time tn = nθ

is given by exp[−=(ωh)tn] = exp[−=(ω∗h)π(utn/`)] = exp[−=(ω∗h)πCn]. This meansthat should =(ω∗h) 6= 0 the amplification at any given time is independent of the timestep θ but dependent on the space discretization `. Thus we present 1d plots for thefollowing:

• Plot of <(ω∗h) vs ξ∗. The departure wavenumber ξ∗d is marked such that ∀ ξ∗ 6ξ∗d we have |<(ω∗ −ω∗h)| 6 0.001 (Figure 1).

• Amplification plots using C = 0.1 and at times θ, 2θ, 100θ, 200θ, and 300θ sec(Figure 2).

1.3.3.2 Fully discrete case

For the fully discrete case the time integration schemes considered are: the backwardEuler, Crank-Nicholson and BDF2. The frequency is now a function of both ξ∗ and C,i. e.ω∗h := ω∗h(ξ

∗,C). The amplification at time tn = nθ is given by exp[−=(ω∗h)π(utn/`)];the same as for the semi-discrete case except for the fact that ω∗h is now dependenton C also. Contour plots are presented for the following,

• log10(|<(ω∗ −ω∗h)|) vs. (ξ∗,C). The contour of values −5,−4,−3,−2,−1 areshown (Figures 3, 5 and 7).

• log10(=(ω∗h)) vs. (ξ∗,C). The contour of values −3,−2.5,−2,−1.5,−1,−0.5, 0

are shown (Figures 4, 6 and 8).

Only the contour plots for the normalized group velocity, i. e.∂ω∗h/∂ξ∗ vs (ξ∗,C),

for the Crank-Nicholson scheme have been presented (Figure 9). The original groupvelocity can be recovered as follows: ∂ωh/∂ξ = u ∂ω∗h/∂ξ

∗.

1.3.4 Discussion

It can be seen from the DDR plots that every discrete model and also the semi-discretemodel of the continuous problem diverges from the exact dispersion relation beyonda certain wave-number, here ξ∗d (Figures 1, 3, 5 and 7). For the semi-discrete case, ξ∗dis marked in the plots and for the fully discrete case ξ∗d is a contour line given by thevalue −3. ξ∗d is greater when we use a consistent mass matrix for the transient termsin the formulation. Thus, one should expect better phase fidelity over a wider rangeof wave numbers using a consistent mass matrix. The gain in the value of ξ∗d fromlumped T case to the consistent T case gradually decreases as the Courant numberC increases (except for a certain range of the Courant number C for the FIC/SUPGmethod).

We now examine the differences in the DDR plots between the Galerkin methodand the DDR plots of the FIC/SUPG and FIC_RC/OSS methods. It is interesting thatthe stabilization terms introduced by the FIC_RC/OSS method do not alter much thelocation of the phase departure wave-number ξ∗d. On the other hand, the stabiliza-tion terms introduced by the FIC/SUPG method contribute to the T matrix (Table 1).


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

0.550.6

0.650.7

0.750.8

0.850.9

0.951

1.051.1

1.151.2

1.251.3

1.351.4

1.451.5

1.551.6

1.651.7

1.751.8

1.851.9

1.952

ξd* = 0.085

Normalized Wavenumber ξ*

Nor

mal

ized

Fre

quen

cy ω

*

Semi−Discrete, Galerkin, Lumped

ω*

Re(ωh* )

Im(ωh* )

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

0.550.6

0.650.7

0.750.8

0.850.9

0.951

1.051.1

1.151.2

1.251.3

1.351.4

1.451.5

1.551.6

1.651.7

1.751.8

1.851.9

1.952

ξd* = 0.279


Nor

mal

ized

Fre

quen

cy ω

*

Semi−Discrete, Galerkin, Consistent

ω*

Re(ωh* )

Im(ωh* )

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

0.550.6

0.650.7

0.750.8

0.850.9

0.951

1.051.1

1.151.2

1.251.3

1.351.4

1.451.5

1.551.6

1.651.7

1.751.8

1.851.9

1.952

ξd* = 0.086


Nor

mal

ized

Fre

quen

cy ω

*

Semi−Discrete, FIC/SUPG, Lumped

ω*

Re(ωh* )

Im(ωh* )

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

0.550.6

0.650.7

0.750.8

0.850.9

0.951

1.051.1

1.151.2

1.251.3

1.351.4

1.451.5

1.551.6

1.651.7

1.751.8

1.851.9

1.952

ξd* = 0.236


Nor

mal

ized

Fre

quen

cy ω

*

Semi−Discrete, FIC/SUPG, Consistent

ω*

Re(ωh* )

Im(ωh* )

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

0.550.6

0.650.7

0.750.8

0.850.9

0.951

1.051.1

1.151.2

1.251.3

1.351.4

1.451.5

1.551.6

1.651.7

1.751.8

1.851.9

1.952

ξd* = 0.085


Nor

mal

ized

Fre

quen

cy ω

*

Semi−Discrete, FIC_RC/OSS, Lumped

ω*

Re(ωh* )

Im(ωh* )

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

0.550.6

0.650.7

0.750.8

0.850.9

0.951

1.051.1

1.151.2

1.251.3

1.351.4

1.451.5

1.551.6

1.651.7

1.751.8

1.851.9

1.952

ξd* = 0.279


Nor

mal

ized

Fre

quen

cy ω

*

Semi−Discrete, FIC_RC/OSS, Consistent

ω*

Re(ωh* )

Im(ωh* )

Figure 1: Plot of ω∗h vs ξ∗ for the semi-discrete problem. Frequencies ω∗ and ω∗h correspondto the continuous and discretized problems respectively. α = 1.0 is used.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

Normalized Angular Wavenumber ξ*

Am

plifi

catio

n

Semi−Discrete, FIC/SUPG, Lumped

time = θtime = 2θtime = 100θtime = 200θtime = 300θ

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05


Am

plifi

catio

n

Semi−Discrete, FIC/SUPG, Consistent


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05


Am

plifi

catio

n

Semi−Discrete, FIC_RC/OSS, Lumped


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05


Am

plifi

catio

n

Semi−Discrete, FIC_RC/OSS, Consistent


Figure 2: Amplification plots of the semi-discrete problem for the FIC/SUPG andFIC_RC/OSS methods. α = 1.0 and C = 0.1 are used. The amplification for theGalerkin method is not shown here as it is equal to 1


−5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−2

−2

−2

−2

−2

−2

−1

−1

−1

−1

−1

−1

Cou

rant

Num

ber

C


Backward Euler, Galerkin, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1 −5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−2

−2

−2

−2

−2

−2

−1

−1

−1

−1

−1−

1

Cou

rant

Num

ber

C


Backward Euler, Galerkin, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−2

−2

−2

−2

−2

−2

−1

−1

−1

−1

−1−1

Cou

rant

Num

ber

C


Backward Euler, FIC/SUPG, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1 −5

−5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4−4

−4

−4

−4

−3

−3

−3

−3

−3

−3 −3−3

−3−3

−3

−2

−2

−2

−2

−2

−2

−2

−2−2

−2 −2

−2

−1−1

−1

−1

−1

−1

−1−1 −1

Cou

rant

Num

ber

C


Backward Euler, FIC/SUPG, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−2

−2

−2

−2

−2

−2

−1

−1

−1

−1

−1

−1

Cou

rant

Num

ber

C


Backward Euler, FIC_RC/OSS, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1 −5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3−3

−3

−2

−2

−2−2

−2

−2−

1−

1−1

−1

−1

−1

Cou

rant

Num

ber

C


Backward Euler, FIC_RC/OSS, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Figure 3: Contour plot of log10[<(|ω∗h−ω∗|)] for the backward Euler scheme. α = 1.0 is used.


−3

−3

−3

−3 −3−

3−

3−

3

−2.5

−2.5

−2.5

−2.5

−2.5

−2.

5−

2.5

−2

−2

−2

−2

−2

−2

−2

−1.5

−1.5

−1.5−1

.5

−1.

5

−1

−1

Cou

rant

Num

ber

C


Backward Euler, Galerkin, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1 −3

−3

−3

−3 −3

−3

−3

−3

−2.5

−2.5

−2.5

−2.5 −2.5

−2.

5−

2.5

−2

−2

−2−2

−2

−2

−2

−1.5

−1.5

−1.5

−1.5

−1.5

−1.

5

−1

−1

−1

−1

−1

Cou

rant

Num

ber

C


Backward Euler, Galerkin, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−3

−3

−3−

2.5−

2.5−2.5

−2

−2

−2−

1.5−

1.5

−1.5−

1

−1

−1

−0.5

−0.

5

−0.5

Cou

rant

Num

ber

C


Backward Euler, FIC/SUPG, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1−

3−

3−

3−

2.5−

2.5

−2.5

−2

−2

−2

−1.5

−1.5

−1.5−

1

−1

−1

−0.

5

−0.5

−0.5

0

0Cou

rant

Num

ber

C


Backward Euler, FIC/SUPG, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−3

−3

−3−

2.5−

2.5

−2.5−

2−

2

−2−

1.5−

1.5

−1.5

−1

−1

−1

−0.

5−

0.5

−0.5

Cou

rant

Num

ber

C


Backward Euler, FIC_RC/OSS, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−3

−3

−3−

2.5−

2.5−2.5

−2

−2

−2−

1.5−

1.5

−1.5−

1−

1

−1

−0.5

−0.

5−

0.5

0

0Cou

rant

Num

ber

C


Backward Euler, FIC_RC/OSS, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Figure 4: Contour plot of log10[=(ω∗h)] for the backward Euler scheme. α = 1.0 is used.


−5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−2

−2

−2

−2

−2

−2

−1

−1

−1

−1

−1

−1

Cou

rant

Num

ber

C


Crank−Nicholson, Galerkin, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−2

−2−2

−2

−2−

2

−1−1

−1−

1−

1−

1

Cou

rant

Num

ber

C


Crank−Nicholson, Galerkin, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−2

−2

−2

−2

−2

−2

−1

−1

−1

−1

−1

−1

Cou

rant

Num

ber

C


Crank−Nicholson, FIC/SUPG, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1 −5

−5

−5

−5

−5 −5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−3

−3

−3

−3

−3

−3

−3

−3

−3

−3

−2

−2

−2

−2

−2

−2

−2

−2

−2

−2

−2

−2

−2

−2

−2−2

−2−1

−1−1

−1

−1

−1

−1

−1

−1

−1

Cou

rant

Num

ber

C


Crank−Nicholson, FIC/SUPG, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−2

−2

−2

−2

−2

−2

−1

−1

−1

−1

−1

−1

Cou

rant

Num

ber

C


Crank−Nicholson, FIC_RC/OSS, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3−3

−3−

3

−3−

3

−3

−3−3

−3

−3

−3

−2

−2−2

−2

−2−

2

−2−

2

−2

−2

−2 −2

−2

−2

−2

−2−

1−1

−1−1

−1

−1

−1

−1

−1

Cou

rant

Num

ber

C


Crank−Nicholson, FIC_RC/OSS, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Figure 5: Contour plot of log10[<(|ω∗h −ω∗|)] for the Crank-Nicholson scheme. α = 1.0 isused.


−3

−3

−3

−2.

5−

2.5

−2.

5

−2

−2

−2

−1.

5−

1.5

−1.

5

−1

−1

−1

−0.

5−

0.5

−0.5

0

Cou

rant

Num

ber

C



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−3

−3

−3

−2.

5−

2.5

−2.

5

−2

−2

−2

−1.

5−

1.5

−1.

5

−1

−1

−1

−0.

5

−0.5

−0.5

0

0

Cou

rant

Num

ber

C



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−3

−3

−3

−2.

5−

2.5

−2.

5

−2

−2

−2

−1.

5−

1.5

−1.

5

−1

−1

−1

−0.

5−

0.5

−0.

5

0

Cou

rant

Num

ber

C



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−3

−3

−3

−2.

5−

2.5

−2.

5

−2

−2

−2

−1.

5−

1.5

−1.

5

−1

−1

−1

−0.

5−

0.5

−0.5

−0.5

0

0

0.5Cou

rant

Num

ber

C



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Figure 6: Contour plot of log10[=(ω∗h)]for the Crank-Nicholson scheme. α = 1.0 is used. The

plots for the Galerkin method is not shown here as =(ω∗h) = 0


−5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−2

−2

−2

−2

−2

−2

−1

−1

−1

−1

−1

−1

Cou

rant

Num

ber

C


BDF2, Galerkin, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1 −5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−2

−2

−2

−2

−2

−2−1

−1

−1

−1

−1−

1

Cou

rant

Num

ber

C


BDF2, Galerkin, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−2

−2

−2

−2

−2

−2

−1

−1

−1

−1

−1

−1

Cou

rant

Num

ber

C


BDF2, FIC/SUPG, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1 −5

−5

−5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−4 −4

−4

−4

−3

−3

−3

−3

−3 −3

−3−3

−3−3

−3

−3

−3

−3

−2

−2

−2

−2

−2 −2 −2

−2

−2−2

−2−2 −2

−2

−1

−1

−1

−1

−1

−1

−1

−1

−1 −1

Cou

rant

Num

ber

C


BDF2, FIC/SUPG, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−2

−2

−2

−2

−2

−2

−1

−1

−1

−1

−1

−1

Cou

rant

Num

ber

C


BDF2, FIC_RC/OSS, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1 −5

−5

−5

−5

−5

−5

−4

−4

−4

−4

−4

−4

−3

−3

−3

−3

−3

−3

−2

−2

−2

−2

−2

−2−1

−1

−1

−1

−1

−1

−1−1

−1

Cou

rant

Num

ber

C


BDF2, FIC_RC/OSS, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Figure 7: Contour plot of log10[<(|ω∗h −ω∗|)] for the BDF2 scheme. α = 1.0 is used.


−6

−6

−6

−6 −6−6

−6

−5.5

−5.5

−5.5−5.5

−5.5

−5.

5−

5.5

−5

−5

−5

−5−5

−5

−5

−4.5

−4.5

−4.5

−4.5

−4.5

−4.

5−

4.5

−4

−4

−4

−4

−4

−4

−3.5

−3.5

−3.5

−3.5

−3.5

−3.

5

−3

−3

−3

−3

−3

−2.5

−2.5

−2.5

−2.5

−2

−2

−2

Cou

rant

Num

ber

C


BDF2, Galerkin, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−6

−6

−6

−6 −6

−6

−6

−6

−5.5

−5.5

−5.5

−5.5−5.5

−5.

5−

5.5

−5

−5

−5

−5

−5−

5−

5−4.5

−4.5

−4.5−4.5

−4.5

−4.

5−

4.5

−4

−4

−4

−4−4

−4

−4

−3.5

−3.5

−3.5−3.5

−3.5

−3.

5−

3.5

−3

−3

−3

−3

−3−

3

−2.5

−2.5

−2.5

−2.5

−2.

5−

2.5

−2

−2

−2

−2

−2

−1.5

−1.5

−1.5

Cou

rant

Num

ber

C


BDF2, Galerkin, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−3

−3

−3

−2.5

−2.5

−2.5

−2

−2

−2

−1.5

−1.5

−1.

5

−1

−1

−1

−0.

5

−0.5

−0.5

Cou

rant

Num

ber

C


BDF2, FIC/SUPG, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1 −3

−3

−3

−2.5

−2.5

−2.5

−2

−2

−2

−1.5

−1.5

−1.

5

−1

−1

−1

−0.5

−0.5

−0.5

0

0Cou

rant

Num

ber

C


BDF2, FIC/SUPG, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−3

−3

−3

−2.5

−2.5

−2.5

−2

−2

−2

−1.5

−1.5

−1.

5

−1

−1

−1

−0.

5

−0.5

−0.5

Cou

rant

Num

ber

C


BDF2, FIC_RC/OSS, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−3

−3

−3

−2.5

−2.5

−2.5

−2

−2

−2

−1.5

−1.5

−1.

5

−1

−1

−1

−0.

5

−0.5

−0.5

0

0

Cou

rant

Num

ber

C


BDF2, FIC_RC/OSS, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Figure 8: Contour plot of log10[=(ω∗h)] for the BDF2 scheme. α = 1.0 is used.


−0.

99−

0.99

−0.

99−

0.99

−0.

99

−0.

95−

0.95

−0.

95−

0.95

−0.

95

−0.

9−

0.9

−0.

9−

0.9

−0.

9

−0.

8−

0.8

−0.

8−

0.8

−0.

8

−0.

7−

0.7

−0.

7−

0.7

−0.

7

−0.

6−

0.6

−0.

6−

0.6

−0.

6

−0.

5−

0.5

−0.

5−

0.5

−0.

5

−0.

4−

0.4

−0.

4−

0.4

−0.

4

−0.

3−

0.3

−0.

3−

0.3

−0.

3

−0.

2−

0.2

−0.

2−

0.2

−0.

2

−0.

1−

0.1

−0.

1−

0.1

−0.

1

00

00

00

0.10.1

0.10.1

0.10.1

0.20.2

0.20.2

0.20.2

0.30.3

0.30.3

0.30.3

0.40.4

0.40.4

0.40.4

0.50.5

0.50.5

0.50.5

0.60.6

0.60.6

0.60.6

0.70.7

0.70.7

0.70.7

0.80.8

0.80.8

0.80.8

0.90.9

0.90.9

0.90.9

0.950.95

0.950.95

0.95

0.990.99

0.990.99

0.99

Cou

rant

Num

ber

C


Crank−Nicholson, Galerkin, Lumped

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−2.

99−

2.99

−2.

99−

2.99

−2.

99−

2.95

−2.

95−

2.95

−2.

95−

2.95

−2.

5−

2.5

−2.

5−

2.5

−2.

5

−2

−2

−2

−2

−2

−2

−1.

5−

1.5

−1.

5−

1.5

−1.

5

−1

−1

−1

−1

−1

−1

−0.

5−

0.5

−0.

5−

0.5

−0.

5

00

00

00

0.10.1

0.10.1

0.10.1

0.20.2

0.20.2

0.20.2

0.30.3

0.30.3

0.30.3

0.40.4

0.40.4

0.40.4

0.50.5

0.50.5

0.50.5

0.60.6

0.60.6

0.60.6

0.70.7

0.70.7

0.70.7

0.8

0.8

0.80.8

0.80.8

0.90.9

0.9

0.90.9

0.9

0.950.95

0.95

0.95

0.950.95

0.990.99

0.990.99

0.990.99

Cou

rant

Num

ber

C


Crank−Nicholson, Galerkin, Consistent

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−2

−2

−2

−2

−2−2

−1.8

−1.8

−1.8

−1.

8−1

.8−1

.8−

1.6−

1.6

−1.

6−

1.6

−1.6

−1.6

−1.

4−

1.4

−1.

4−

1.4

−1.4

−1.4

−1.

2−

1.2

−1.

2−

1.2

−1.2

−1.2

−1

−1

−1

−1−1

−1

−0.

8−

0.8

−0.

8−0

.8−0

.8−0.8

−0.

6−

0.6

−0.

6−0

.6−0

.6

−0.6

−0.

4−

0.4

−0.

4−0

.4

−0.4

−0.4

−0.

2−

0.2

−0.

2−0

.2

−0.2

−0.2

00

00

0

0

0

0.1

0.1

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.3

0.3

0.3

0.3

0.3

0.30.4

0.4

0.4

0.4

0.4

0.4

0.50.5

0.50.5

0.50.5

0.60.6

0.60.6

0.60.6

0.70.7

0.70.7

0.70.7

0.80.8

0.80.8

0.80.8

0.90.9

0.90.9

0.90.9

0.950.95

0.950.95

0.95

0.990.99

0.990.99

0.99

Cou

rant

Num

ber

C



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−12.5

−10

−10

−7.

5−7

.5

−5

−5

−2.5

−2.5

00

0

0.5

0.5

0.91 0.91

0.91

0.91

0.91

0.93

0.930.93

0.93

0.93

0.93

0.95

0.95

0.95

0.95

0.95

0.95

0.95

0.970.97

0.97

0.97

0.97

0.97

0.97

0.97

0.990.99

0.99

0.99

0.99

0.99

0.99

0.99

0.99

1

1

1

1

1

1

1 11

1

1.01

1.01

1.01

1.01

1.01

1.01

1.01

1.01

1.05

1.05

1.05

1.05

1.05

1.05

1.05

1.051.

11.

1

1.1

1.1

1.1

1.1

1.1

1.1

1.2

1.2

1.2

1.2

1.2

1.2

1.2

1.2

1.3

1.3

1.3

1.3

1.3

1.3

1.3

1.3

1.4

1.4

1.4

1.4

1.4

1.4

1.4

1.4

1.4

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.6

1.6

1.6

1.6

1.6

1.6

1.6

1.6

1.6

1.7

1.7

1.7

1.7

1.7

1.7

1.7

1.7

1.7

1.8

1.8

1.8

1.8

1.8 1.8

2

2

2

2

2

2.5

2.5

2.5

2.5

3

3

3

3.5

3.5

3.5

4

4

4Cou

rant

Num

ber

C



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1−0.99

−0.99−

0.99

−0.

99−0

.99

−0.9

9−

0.95−

0.95−

0.95

−0.

95−0

.95

−0.95

−0.9

−0.9

−0.9

−0.

9−0

.9

−0.9

−0.

8−

0.8

−0.

8−

0.8

−0.8

−0.8

−0.

7−

0.7

−0.

7−0

.7−0

.7

−0.7

−0.

6−

0.6

−0.6

−0.6

−0.6

−0.6

−0.

5−

0.5

−0.5

−0.5

−0.5

−0.5

−0.

4−

0.4

−0.4

−0.4

−0.4

−0.4

−0.

3−

0.3

−0.3

−0.3

−0.3

−0.3

−0.

2−

0.2

−0.2

−0.2

−0.2

−0.2

−0.

1−

0.1

−0.

1−0

.1

−0.1

−0.1

00

00

0

0

0

0.1

0.1

0.1

0.1

0.1

0.1 0.1

0.1

0.2

0.2

0.2

0.2

0.2

0.2 0.2 0.20.3

0.3

0.3

0.3

0.3

0.3

0.30.3

0.40.4

0.40.4

0.40.4

0.40.50.5

0.50.5

0.50.5

0.50.60.6

0.60.6

0.60.6

0.6

0.70.7

0.70.7

0.70.7

0.80.8

0.80.8

0.80.8

0.90.9

0.90.9

0.90.9

0.950.95

0.950.95

0.95

0.990.99

0.990.99

0.99

Cou

rant

Num

ber

C



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

−3

−3−

2.5

−2.5

−2

−2

−1.

5−1

.5

−1

−1

−1

−0.

5

−0.5

−0.5

0

0

0

0.1

0.1

0.1

0.2

0.2

0.2

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.5

0.5

0.6

0.6

0.6

0.6

0.6

0.7

0.7

0.70.7

0.7

0.7

0.8

0.8

0.8

0.8

0.80.8

0.8

0.8

0.90.9

0.9

0.90.9

0.9

0.9

0.9

0.9

0.9

0.9 0.9

0.9

0.9

0.950.95

0.95

0.950.95

0.95

0.95

0.950.95

0.95

0.95 0.95

0.95

0.95

0.990.99

0.990.99

0.99

0.99

0.990.99

0.99

0.99 0.99

0.99

0.99

1.2

1.21.2

1.2

1.2

1.2

1.2

1.2

1.5

1.51.5

1.5

1.5

1.5

1.5

1.5

2

2

2

2

22

22

2.5

2.5

2.5

2.52.5

2.52.5

3

3

3

3

3

3

3.5

3.5

3.5

3.5

3.5

4

44

4

4Cou

rant

Num

ber

C



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Figure 9: Real part of the normalized group velocity ∂ω∗h/∂ξ∗ vs. ξ∗ for the Crank Nicholson

scheme; α = 1.0 is used.


Although for higher Courant number C the effect on the location of ξ∗d is negligible,significant alterations in ξ∗d are found for a lower C and a consistent T matrix (Figures3, 5 and 7). When T is lumped, all the methods give similar patterns for ξ∗d indicatingthat stabilization terms have no role to play in the improvement of the DDRs.

Next, we examine the effect of the variation in C on the <(ω∗h) vs ξ∗ relation for theFIC and SUPG methods with consistent T matrix and the Crank Nicholson scheme.Again, we choose ξ∗d using the criterion: ∀ ξ∗ 6 ξ∗d we have |<(ω∗ −ω∗h)| 6 0.001.In Figure 5 this corresponds to the contour line for the value −3. It can been seenin the semi-discrete case that <(ω∗h) > <(ω∗) for lower wavenumbers (Figure 1). Inaddition, the error <(ω∗h −ω∗) first increases and later decreases. This behavior isexhibited in the fully discrete case too. Thus, there will be multiple contours of thesame value in the log10(|ω

∗h −ω∗|) vs (ξ∗,C) contour plots. In this case ξ∗d is the

smallest ξ∗ on the contours. For instance, choosing C = 0.4 for the FIC and SUPGmethods we find ξ∗d ≈ 0.35, where as for the Galerkin, FIC_RC and OSS methods wefind ξ∗d < 0.2 (Figure 5). Thus, there is a significant improvement in the DDR for theFIC and SUPG methods for this value of C. It is interesting to note that for C 6 0.1the variation of DDR with C is insignificant. Similar conclusion can be made for thebackward Euler and BDF2 schemes from their respective DDR plots.

It can be seen from the amplification plots (Figures 2, 4, 6 and 8) that for the samevalue of α (here α = 1.0) and using a consistent T matrix, the damping associatedwith the FIC/SUPG method is relatively less than that of the FIC_RC/OSS method,though the gain is not significant for low wavenumbers. On the other hand, usinga lumped T matrix the difference in the amplification associated with FIC/SUPGand FIC_RC/OSS methods is insignificant. It can also be seen that unlike the notabledifferences in the location of ξ∗d, the differences in the amplification due to lumpingthe T matrix is insignificant.

An important aspect is the effect of group velocity. This is more evident when thetransported function is periodic and resembles a sinusoid wave train. For such func-tions, the Fourier transform is a narrow peak concentrated around the characteristicangular wave number of the function. For such problems the effect of the group veloc-ity is more significant than that of phase velocity. If the DDR predicts a deviation inthe group velocity then the wave train travels at that deviant velocity (Example 1.6.3).

1.4 stabilization parameters

In this section, the optimal expressions of the stabilization parameters for the FIC_RCmethod in 1d and on a uniform mesh are proposed. We consider the steady-state formof the discrete problem (1.17) for the sourceless case (f = 0). The equation stencil foran interior node j is as follows:

(u

2)[φj+1 −φj−1] − (k+

uh

2)[φj+1 − 2φj +φj−1

`] + (

uh

8)[φj+2 − 2φj +φj−2

`] = 0

(1.27)

The analytical solution of the steady-state form of problem (1.17) in the 1d spacewith only Dirichlet boundary conditions and source f = 0 is,

φ(x) = φpl + (φpr −φpl )

[exp[ux/k] − 1exp[uL/k] − 1

](1.28)


−20 −15 −10 −5 0 5 10 15 20−2

−1.75

−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

α

γ

Plot of α = −sinh−2(γ)[coth(γ) − γ−1]

Figure 10: Optimal expression of α for the FIC_RC/OSS method. Plot of α vs. γ for the interiornodes.

Figure 11: Node stencil for the 1d problem.

where L is the length of the 1d domain and φpl ,φpr are the prescribed values of φ atthe left and right ends of the domain respectively. We now express the characteristiclength in terms of the element size as h = α `. The optimal value of the stabilizationparameter α is found by substituting the analytical solution into the stencil given inEq.(1.27). This leads to the following expression of α for the interior nodes.

α =−1

sinh2(γ)

[coth(γ) −

1

γ

](1.29)

where, γ is the element Peclet number given by γ = u`2k . The plot of the parameter α vs.

γ is shown in Figure 10.It can be seen that α ≈ 0 for γ > 3. So for high Peclet numbers the formulation

suggests to use very small values of α. That is for large values of γ the formulationbreaks down into the standard Galerkin method without stabilization and one canexpect spurious oscillations in the numerical solution. The clues to reason out thisbehavior can be found by examining the assembly of the linear system.

The 1d problem is discretized by linear elements and the final form of the finiteelement assembly is examined. For simplicity, the nodes are numbered from left toright as shown in Figure 11. For the interior nodes, the equation stencil is as given byEq.(1.27). For the left penultimate boundary node, here node 2 as per the numberingscheme Figure 11, we find the following stencil:

(u

2)[φ3 −φ1] − (k+

uh

2)[φ3 − 2φ2 +φ1

`] + (

uh

8)[φ4 − 3φ2 + 2φ1

`] = 0 (1.30)


−20 −15 −10 −5 0 5 10 15 20−2

−1.75

−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2α 2

γ

Plot of α2 = 4*(2−e(2γ))−1[coth(γ) − γ−1]

−20 −15 −10 −5 0 5 10 15 20−2

−1.75

−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

α n

γ

Plot of αn = 4*(2−e(−2γ))−1[coth(γ) − γ−1]

Figure 12: Plot of optimal α (for the penultimate boundary nodes) vs. γ for the FIC_RC/OSSmethod.

For the right penultimate boundary node, i.e the node n in Figure 11, we find thefollowing stencil:

(u

2)[φn+1 −φn−1] − (k+

uh

2)[φn+1 − 2φn +φn−1

`]

+ (uh

8)[2φn+1 − 3φn +φn−2

`] = 0 (1.31)

Thus, we see that there is a rearrangement of the equation stencils for the nodes thatlie next to the boundary. The stencils for the boundary nodes also gets rearranged,but we do not consider them here as the focus here is to deal problems with Dirichletboundary conditions. The deviation of the nodal equations for the penultimate nodesfrom the interior nodes is responsible for the spurious oscillations. It has been shownusing a spectral analysis framework in [163, 164] that the asymmetry in the stencilsbrings about anti-diffusion even when they are used with central difference schemesand their effect is not localized, thus being responsible for spurious numerical oscil-lations. A simpler explanation would be that the rearrangement of the stencils nearthe boundary require different expressions for the stabilization parameter for thosenodes.

The optimal values of the stabilization parameters for these penultimate nodes on auniform mesh are as follows. Eqs.(1.32a) and (1.32b) correspond to the optimal valuesfor the nodes 2 and n respectively. Figure 12 illustrates the variation of α with respectto γ.

α2 =4

[2− e2γ]

[coth(γ) −

1

γ

](1.32a)

αn =4

[2− e−2γ]

[coth(γ) −

1

γ

](1.32b)

An alternative to nodal stabilization parameters is to find optimal values of the sta-bilization parameters for the elements adjacent to the boundary. For elements laying


−20 −15 −10 −5 0 5 10 15 20−4

−3.75−3.5

−3.25−3

−2.75−2.5

−2.25−2

−1.75−1.5

−1.25−1

−0.75−0.5

−0.250

0.250.5

0.751

1.251.5

1.752

2.252.5

2.753

3.253.5

3.754

α(1)

γ

Plot of α(1) = 4*(1−e(2γ))−1[coth(γ) − γ−1]

−20 −15 −10 −5 0 5 10 15 20−4

−3.75−3.5

−3.25−3

−2.75−2.5

−2.25−2

−1.75−1.5

−1.25−1

−0.75−0.5

−0.250

0.250.5

0.751

1.251.5

1.752

2.252.5

2.753

3.253.5

3.754

α(n)

γ

Plot of α(n) = 4*(1−e(−2γ))−1[coth(γ) − γ−1]

Figure 13: Plot of optimal α (for the elements adjacent to the boundary) vs. γ for theFIC_RC/OSS method.

in the domain interior we use the value given by Eq.(1.29). Denoting the stabilizationparameters for elements 1, n and in the domain interior as α(1), α(n) and α respec-tively, we find the following equation stencils.

Node 2:

(u

2)[φ3 −φ1] − (

k

`)[φ3 − 2φ2 +φ1]

− (u

2)[αφ3 − (α(1) +α)φ2 +α

(1)φ1]

+ (u

8)[αφ4 + (α−α(1))φ3 − (α+ 2α(1))φ2 + (3α(1) −α)φ1] = 0 (1.33)

Node n:

(u

2)[φn+1 −φn−1] − (

k

`)[φn+1 − 2φn +φn−1]

− (u

2)[α(n)φn+1 − (α(n) +α)φn +αφn−1]

+ (u

8)[(3α(n) −α)φn+1 − (α+ 2α(n))φn + (α−α(n))φn−1 +αφn−2] = 0

(1.34)

The optimal values for α(1) and α(n) are given by the Eqs.(1.35a) and (1.35b) re-spectively. Figure 13 illustrates the variation of α with respect to γ.

α(1) =4

[1− e2γ]

[coth(γ) −

1

γ

](1.35a)

α(n) =4

[1− e−2γ]

[coth(γ) −

1

γ

](1.35b)

Remark: It is important to note that α is a signed parameter. For the domain interior,α is negative for positive values of γ! As the optimal α on a uniform grid for theFIC_RC/OSS method is close to zero everywhere within the domain interior (Figure

1.5 Discrete Maximum Principle 29

10) we do not introduce any artificial (unnatural) damping anywhere within the do-main. The effect of boundary shock layers can also be captured efficiently by the newexpressions of α at the boundary-adjacent elements. Thus, we see that in the tran-sient mode the performance of the FIC_RC/OSS method using the proposed optimalα with boundary correction is similar to the standard Galerkin method. The formermethod also gives nodally exact solutions (on uniform grids) in the steady state modeunlike the spurious global oscillations produced by the later method. Of course, allthese features are favorable addenda only if the bandwidth of the amplitude spectraof the transported function lies within the range of the phase departure wave number.

1.5 discrete maximum principle

On further examination, the optimal expression (on uniform grids in 1d) of the sta-bilization parameter for the FIC_RC/OSS method illustrates certain subtle featuresrelated to the satisfaction of a discrete maximum principle (DMP). If the DMP holdsfor any numerical method then the maximum component in the modulus of the so-lution vector is bounded above by the maximum component in the modulus of theboundary data. It is well known that a sufficient condition for the satisfaction of theDMP is that the system matrix be an M-matrix or a matrix of ’positive type’. We referto [17, 40] for the definition of the latter matrices.

Note that for Peclet numbers greater than one, the system matrix (see Eq.(1.27))associated with the FIC_RC/OSS method can never be cast into the form of an M-matrix or a matrix of ’positive type’. However, using the proposed optimal expressionfor α we get nodally exact solutions on a uniform mesh in 1d. It is remarkable thatin this case the system matrix is not even a monotone matrix whose definition can befound in [17, 40]. Nevertheless, the FIC_RC/OSS system matrix obtained using theoptimal α verifies the necessary and sufficient condition given in [186] for the DMPto hold. Unfortunately, this necessary and sufficient condition for a DMP to hold isdifficult to identify a priori and thus poising a strategical difficulty in the design ofdiscontinuity/shock capturing methods.

1.6 numerical examples

1.6.1 Example 1

In this example, we study the effect of the stabilization introduced by FIC/SUPG andFIC_RC/OSS methods for transient problems. We study the pure convection prob-lem primarily for two reasons. First, the problem becomes simple as the dispersioneffect of natural diffusion is sidelined, thus allowing us to study the effects of thediffusion introduced by the stabilization methods. Next, the convection dominatedproblem is the primary concern of stabilization methods. The domain of interest isx ∈ [0, 10]. The problem data are k = 1× 10−30, u = 1.0, time increment θ = 0.01and the 1d space is discretized by 100 linear elements with uniform mesh size. Thus,` = 0.1. The Peclet number for this problem is γ = 5× 1028 (γ ≈ ∞). The Courantnumber is C = 0.1. We have also chosen α = 1.0, the optimal value for SUPG in thiscase, throughout the simulations. This choice is made just to study the effect of theartificial diffusion introduced by the FIC_RC/OSS method within the interior of the


domain using a non-optimal value for α. Recall that the optimal value of α for theFIC_RC/OSS method in this case is ≈ 0, thus behaving just like the standard Galerkinmethod in the domain interiors. The discretization in time is done by the followingschemes: Crank-Nicholson, backward Euler and BDF2 schemes.

Figure 14 shows the results obtained when a narrow Gaussian pulse centered atx = 3.0 is taken as the initial solution. The equation for the initial solution is φo(x) =exp[−8(x−3)2]. The numerical solution at time 3s is examined. The solutions obtainedusing a lumped and consistent T matrix in the formulation are also examined. Theidea is to validate the conclusions that can be drawn from the DDR plots. The pulsewidth of the Gaussian function is chosen as to guarantee that the bandwidth of theamplitude spectra of this function is just less than ξconsistentd , the phase departurewave-number using a consistent T matrix. As ξlumpedd 6 ξconsistentd , one shouldexpect incorrect superpositions of the wave trains due to their phase differences fromthe former. This leads to a train of wiggles as seen in the numerical solution when Tis lumped (Figure 14).

It is interesting that the standard Galerkin method without any stabilization and byusing a consistent T matrix yields very accurate results. The other methods using aconsistent T create a slight bump at the foot of the Gaussian bell. This is because of thedamping associated with those methods for the significant wavenumbers (ξ∗ 6 ξ∗d).We also notice that using a consistent T, the damping associated with the FIC/SUPGmethod is less than that for the FIC_RC/OSS method. The differences are insignificantfor the T-lumped case.

1.6.2 Example 2

In this example, we illustrate the effect of the variation of the Courant number onthe DDR. The problem data are the same as in Example 1.6.1. Time integration isperformed by the Crank-Nicholson scheme. Two initial functions are considered : anarrow Gaussian pulse centered at x = 3.0 as defined in Example 1.6.1 and a squarepulse function defined by φo(x) = 1.0 if x ∈ [2, 4] else φo(x) = 0.0. The spectra of thesquare pulse is broad and the bandwidth extends beyond the ξd of all the methodsconsidered here. In other words, in the absence of damping this function will exhibitnumerical dispersion. In this example only the consistent T matrix is used.

First we note that for the higher Courant number (here C = 1.0) the Gaussian pulseexhibit numerical dispersion even when a consistent T matrix is used (Figure 15).As C is reduced to 0.4 and 0.1 we notice that the dispersion errors are minimized.As discussed earlier in §1.3.4 the FIC/SUPG method with C = 0.4 should exhibita better performance over the FIC_RC/OSS method. Unfortunately the gain in theDDR for the higher wavenumbers does not materialize in the simulated results. Thesolutions for the FIC/SUPG and the FIC_RC/OSS methods are nearly the same forboth the initial solutions (Figure 15). This is because all those wavenumbers sufferhigh damping. As =(ω∗h) does not vary with C (Figure 6), Figure 2 may be referredfor the amplification.


0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

ΦGaussian Bell − Crank Nicholson − Lumped − Time: 3.00 s

ExactGalerkinFIC/SUPGFIC_RC/OSS

0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

Φ

Gaussian Bell − Crank Nicholson − Consistent − Time: 3.00 s


0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

Φ

Gaussian Bell − Backward Euler − Lumped − Time: 3.00 s


0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

Φ

Gaussian Bell − Backward Euler − Consistent − Time: 3.00 s


0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

Φ

Gaussian Bell − BDF2 − Lumped − Time: 3.00 s


0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

Φ

Gaussian Bell − BDF2 − Consistent − Time: 3.00 s


Figure 14: Example 1: transport of a Gaussian pulse. Solutions at 3s for the Galerkin, FIC/-SUPG and FIC_RC/OSS methods using lumped (on left) and consistent (on right)T matrix are shown; Time discretization is done by the Crank-Nicholson, backwardEuler and BDF2 schemes. α = 1.0 is used.


0 1 2 3 4 5 6 7 8 9 10−0.3

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1

1.1

x

Φ

Gaussian Bell − Consistent − Courant 0.1 − Time: 3.00 s


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0 1 2 3 4 5 6 7 8 9 10−0.3

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Φ

Square Pulse − Consistent − Courant 0.4 Time: 3.00 s


0 1 2 3 4 5 6 7 8 9 10−0.3

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x

Φ

Square Pulse − Consistent − Courant 1.0 Time: 3.00 s


Figure 15: Example 2: transport of a Gaussian and square pulse. Solutions at 3s for theGalerkin, FIC/SUPG and FIC_RC/OSS methods using a consistent T matrix andfor Courant number 0.1, 0.4 and 1.0; Time discretization performed by Crank-Nicholson scheme. α = 1.0 is used.


1.6.3 Example 3

In this example, we study the transport of a periodic sine wave with angular wave-number ξo = 7.5. The problem data is the same as in Example 1.6.1. The corre-sponding value of ξ∗ = 0.238. Time integration is performed by the Crank-Nicholsonscheme. The idea here is to study the effect of the group velocity in the numericalsimulation. The initial function and the boundary condition prescribed at the leftboundary are as follows:

f(x, 0) = sin(ξox) (1.36)

f(0, t) = sin(uξot) (1.37)

The DDR predicts a group velocity Vlumpedg ≈ 0.75 and Vconsistentg ≈ 1.0 for ξ∗ =0.238 using a lumped and consistent T matrices, respectively (Figure 9). This propertyis exhibited in the numerical solution where we find that the wave train moves at adifferent speed from the one assigned. The results are accurate using a consistent Tmatrix (Figure 16). The damping introduced by the stabilization methods are also inagreement with the one predicted by the DDR of the problem. Using a consistent Tmatrix the amplifications for the FIC/SUPG and the FIC_RC/OSS methods for α = 1.0and the wavenumber ξo = 7.5 are ≈ 0.7 and 0.3, respectively, after 3s (Figure 2). Thecorresponding values for the T-lumped case are ≈ 0.4 and 0.32. The numerical resultsare in agreement with this prediction. We note that the numerical damping associatedwith the FIC/SUPG method is less than that of the FIC_RC/OSS method for the samevalue of α. An interesting result is that when the expressions (Eqs. 1.29,1.35) for αare used for the FIC_RC/OSS method, no damping takes place as it behaves similarto the Galerkin method in the interior domain. The numerical solution in this casecoincides with that shown for the Galerkin method.

1.6.4 Example 4

In this example, we explore the performance of the stabilization parameter α givenby Eqs. (1.29) and (1.35) for the steady state problem using the FIC_RC/OSS method.The domain of interest is x ∈ [0.0, 1.0]. The problem data are k = 0.001, u = 1.0 andf = 1.0. For the ease of notation and further reference we define the following:

αa =

[coth(γ) −

1

γ

]

αb =

4

[1− e2γ]

[coth(γ) −

1

γ

], element 1

4

[1− e−2γ]

[coth(γ) −

1

γ

], element n

−1

sinh2(γ)

[coth(γ) −

1

γ

], else

First, we study the solution on a uniform mesh consisting of 20 linear elements (` =0.05). We consider the cases when f = 0 and f = 1.0. The numerical solutions of the


0 1 2 3 4 5 6 7 8 9 10−1.1

−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.20.30.40.50.60.70.80.9

11.11.21.31.41.5

x

Φ

Sine Wave − Crank Nicholson − Lumped − Time: 1.00 s


0 1 2 3 4 5 6 7 8 9 10−1.1

−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.20.30.40.50.60.70.80.9

11.11.21.31.41.5

x

Φ

Sine Wave − Crank Nicholson − Consistent − Time: 1.00 s


0 1 2 3 4 5 6 7 8 9 10−1.1

−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.20.30.40.50.60.70.80.9

11.11.21.31.41.5

x

Φ

Sine Wave − Crank Nicholson − Lumped − Time: 3.00 s


0 1 2 3 4 5 6 7 8 9 10−1.1

−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.20.30.40.50.60.70.80.9

11.11.21.31.41.5

x

Φ

Sine Wave − Crank Nicholson − Consistent − Time: 3.00 s


Figure 16: Example 3: transport of a sinusoidal wave. Solutions at 1s and 3s for the Galerkin,FIC/SUPG and FIC_RC/OSS methods using a lumped (on left) and consistent (onright) T matrix; Time discretization done by the Crank-Nicholson scheme. α = 1.0is used.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

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x

φUniform Grid, φp(0) = 0.0; φp(1) = 1.0, f = 0.0

FIC/SUPG,αa

FIC_RC/OSS,αa

FIC_RC/OSS,αb

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

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φ

Non−uniform Grid, φp(0) = 0.0; φp(1) = 1.0, f = 0.0

FIC/SUPG,αa

FIC_RC/OSS,αa

FIC_RC/OSS,αb

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

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x

φ

Uniform Grid, φp(0) = 0.0; φp(1) = 0.0, f = 1.0

FIC/SUPG,αa

FIC_RC/OSS,αa

FIC_RC/OSS,αb

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

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0.5

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1

1.1

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x

φ

Non−uniform Grid, φp(0) = 0.0; φp(1) = 0.0, f = 1.0

FIC/SUPG,αa

FIC_RC/OSS,αa

FIC_RC/OSS,αb

Figure 17: Example 4: steady-state solutions on uniform and nonuniform grids for the FIC/-SUPG and FIC_RC/OSS methods. αa := [coth(γ) − γ−1]. αb is evaluated via theEqs. (1.29) and (1.35)

FIC/SUPG method using the stabilization parameter as αa and of the FIC_RC/OSSmethod using αa and αb are presented in Figure 17. We note that the new definitionfor the stabilization parameter αb is optimal for the uniform mesh. The boundarycorrection introduced in Eq.(1.35) also takes effect.

Next, we study the solution on a non-uniform grid consisting of 15 elements. Thenode coordinates of the discrete 1d space are given by x = 0, 0.095, 0.191, 0.2866,0.382, 0.477, 0.573, 0.656, 0.728, 0.789, 0.841, 0.885, 0.922, 0.953, 0.979, 1. We note thatthe solution of the FIC/SUPG method is superior to that obtained by the FIC_RC/OSSmethod. The later method gives the sharp boundary oscillations using the parameterαa (also appears on uniform grids). The solution of the FIC_RC/OSS method usingαb on the non-uniform grid is slightly corrupted with weak node-to-node spuriousoscillations.


1.7 conclusions

A detailed transient analysis of a consistency recovery method (FIC_RC/OSS) withrespect to the Galerkin and FIC/SUPG methods has been done. The discrete disper-sion relations for the above methods using the trapezoidal and BDF2 time integrationschemes are presented. The phase departure wavenumber (ξd) is greater when a con-sistent matrix for the transient terms is used. ξd proportionally influences the rangeof wavenumbers that have a group velocity close to the convection velocity. The DDRplots predict that the gain in the value of ξd from the lumped T case to the consis-tent T case gradually decreases with the Courant number C. An exception to this ison a certain lower range of C for the FIC/SUPG method. Unfortunately, this gain isseldom realized as those higher wavenumbers are damped away. The contour plots oflog[<(|ω∗h −ω∗|)] are very similar for the Galerkin, FIC/SUPG and the FIC_RC/OSSmethods except for the exception mentioned earlier. Neither is the change significantwith the choice of the time integration schemes considered. This suggests that therole of the stabilization terms in the improvement of the DDR is insignificant and theenhancement can be achieved only by virtue of the resolution in space and time. Itis shown that for the same value of the stabilization parameter α, the damping asso-ciated with the FIC_RC/OSS method is slightly greater than that of the FIC/SUPGmethod.

It is shown that, unlike the FIC/SUPG method, the FIC_RC/OSS method intro-duces a certain rearrangement in the equation stencils at nodes on and adjacent to thedomain boundary. Thus using a uniform expression for α will lead to localized os-cillations at the boundary. These oscillations are notable in the convection dominatedcase. When diffusion is at par with convection, the solution has a smooth profile andthese local oscillations are insignificant even though they exist. A proposal for α thatgives optimal results for the steady-state 1d convection diffusion problem on uniformgrids is made for the FIC_RC/OSS method. An interesting result is that when thenew expressions for α are used for the FIC_RC/OSS method, no damping takes placeas α ≈ 0 in the domain interior. The numerical solution in the transient case coincideswith that for the Galerkin method and in the steady state, unlike the Galerkin method,it is stable. Unfortunately, using this new expression for α, dispersion errors when-ever present (for instance, the transport of a square pulse), cannot be controlled andalso the steady state solution has weak node-to-node oscillations on a non-uniformgrid. On the basis of these results, it appears that with respect to the stabilization ofconvection, the numerical performance of the FIC/SUPG method is better as one cancontrol to some extent the dispersive errors and at the same time we can assure thestability of the steady-state solution.

Finally, it can be verified that the FIC_RC/OSS system matrix obtained using theoptimal stabilization parameter (on uniform grids) is neither a monotone matrix nor amatrix of positive type. Although, it verifies the necessary and sufficient condition fora DMP to hold. Unfortunately, it is difficult to identify a priori the matrices that satisfythis necessary and sufficient condition unlike the matrices of ’positive type’ that areeasy to recognize. This poses a strategical difficulty in the design of discontinuitycapturing methods. Due to these difficulties we currently do not prefer the consistencyrecovery method in the stabilization of convection.

What is karma? and what is not karma?are questions that perplex the wisest of men.

— Bhagawad Gita IV:16.

2A H I G H - R E S O L U T I O N P E T R O V– G A L E R K I N M E T H O D I N 1 D

2.1 introduction

A singularly perturbed convection–diffusion–reaction problem is an initial-boundaryvalue problem where the diffusion coefficient may take arbitrarily small values. Thesolution of this problem may exhibit transient and/or exponential boundary layers.In higher dimensions the solution may also exhibit characteristic boundary and/orinterior layers. It is well known that the numerical solution of this problem by theBubnov–Galerkin finite element method (FEM) is prone to exhibit global, Gibbs anddispersive oscillations. The solution of the stationary problem by the above method ex-hibits spurious global oscillations for the convection-dominated cases. The local Gibbsoscillations are exhibited along the characteristic layers for the convection-dominatedcases. For the reaction-dominated cases Gibbs oscillations may be found near theDirichlet boundaries and in the regions where the distributed source term is nonreg-ular. The solution of the transient problem may exhibit dispersive oscillations shouldthe initial solution and/or the distributed source term are nonregular.

In the context of variational formulations and weighted residual methods, con-trol over the global instability has been achieved via the streamline-upwind Petrov–Galerkin (SUPG) [22, 92], Taylor–Galerkin [53], characteristic Galerkin [55, 127], Galerkinleast squares (GLS) [95], bubble functions [12, 18, 19], variational multiscale (VMS)[91, 97], characteristic-based split (CBS) [193] and finite calculus (FIC) based meth-ods [141]. A thorough comparison of some of these methods can be found in [32].Close connections between the VMS method and stabilization via bubble functionswas pointed out in [20]. It was shown that some of the above stabilized methods canbe recovered using the FIC equations via an appropriate definition of the stabilizationparameters [141, 146]. Nevertheless nonregular solutions continue to exhibit the Gibbsand dispersive oscillations.

Several shock-capturing nonlinear Petrov–Galerkin methods were proposed to con-trol the Gibbs oscillations observed across characteristic internal/boundary layers forthe convection-diffusion problem [23, 29, 46, 48–50, 68, 94, 116, 119, 132, 154]. Athorough review, comparison and state of the art of these and several other shock-capturing methods for the convection-diffusion equations, therein named as spuriousoscillations at layers diminishing methods, was done in [114]. Reactive terms were notconsidered in the design of these methods and hence they fail to control the localizedoscillations in the presence of these terms. Exceptions to this are the consistent approx-imate upwind (CAU) method [68], the methods presented in [23, 30] and those thattake the CAU method as the starting point [48–50]. Nevertheless the expressions forthe stabilization parameters therein were never optimized for reactive instability andoften the solutions are over-diffusive in these cases.

37

38 2 a high-resolution petrov–galerkin method in 1d

In the quest to gain reactive stability several methods were built upon the exist-ing frameworks of methods that control Global oscillations. Following the frame-work of the SUPG method linear Petrov–Galerkin methods were proposed for theconvection–diffusion–reaction problem, viz. the DRD [180] and (SU+C)PG [100] meth-ods. Based on the GLS method linear stabilized methods were proposed, viz. theGGLS method [61] for the diffusion–reaction problem and GLSGLS method [80] forthe convection-diffusion-production problem. Within the framework of stabilizationvia bubbles the USFEM method [62] for the diffusion–reaction problem, the improvedUSFEM method [66] and the link cutting bubbles [21] for the convection–diffusion–reaction problem were proposed. Based on the VMS method linear stabilized methodswere proposed for the convection–diffusion–reaction problem, viz. the ASGS method[33], the methods presented in [84, 85] and the SGS-GSGS method [86]. Using theFIC equations a nonlinear method based on a single stabilization parameter wasproposed for the convection–diffusion–reaction problem [152, 155]. This nonlinearmethod, though initially formulated within the Petrov–Galerkin framework, subse-quent modeling of a simplified form for the numerical nonlinear diffusion, deviatedthe method from being residual-based (consistency property is violated). Nodally ex-act Ritz discretizations of the 1d diffusion-absorption/production equations by varia-tional FIC and modified equation methods using a single stabilization parameter werepresented in [59]. Generally the homogeneous steady convection–diffusion–reactionproblem in 1d has two fundamental solutions. Likewise, the characteristic equation as-sociated with linear stabilized methods which result in compact stencils are quadraticand hence have two solutions. Thus in principle using two stabilization parameters(independent of the boundary conditions) linear stabilized methods which result incompact stencils can be designed to be nodally exact in 1d. Following this line sev-eral ‘two-parameter methods’ viz. (SU+C)PG, GLSGLS and SGS-GSGS methods weredesigned to be nodally exact for the stationary problem in 1d.

Control over the dispersive oscillations for the transient convection-diffusion prob-lem via linear Petrov–Galerkin methods were discussed in [99] and using space-timefinite elements in [189]. As for the linear methods, optimizing the expressions of thestabilization parameters to attain monotonicity will lead to solutions that are at mostfirst-order accurate.

Out of the context of variational formulations and weighted residual methods, avast literature exists on the design of high-resolution methods. These methods arecharacterized as algebraic flux correction/limiting methods and are usually developedwith in the framework of finite-difference (FDM) or finite-volume (FVM) models. Werefer to the books [89, 123, 124, 182] for a review of these methods and to the seminalpapers in this field [16, 82, 83, 185, 191]. The book [120] describes the state of the art inthe development of high-resolution schemes based on the Flux-Corrected Transport(FCT) paradigm for unstructured meshes and their generalization to the FEM. Nev-ertheless the use of these schemes were reported to be rather uncommon in spite oftheir enormous potential. We refer to the introduction in [161] that discusses the pop-ularity of methods based on variational formulations and weighted residuals. Thus,as encouraged therein, the quest for the design of high-resolution methods based onvariational/weighted-residual formulations is active to date.

In this chapter we present the design of a FIC-based nonlinear high-resolutionPetrov–Galerkin (HRPG) method for the 1d convection–diffusion–reaction problem.


The prefix ‘high-resolution’ is used here in the sense popularized by Harten, i. e.secondorder accuracy for smooth/regular regimes and good shock-capturing in nonregu-lar regimes. The goal is to design a numerical method within the context of theFIC variational formulation and weighted residuals which is capable of reproduc-ing high-resolution numerical solutions for both the stationary (efficient control ofglobal and Gibbs oscillations as seen in methods [21, 59, 80, 86, 100, 152, 155]) andtransient regimes (efficient control of dispersive oscillations as seen in algebraic fluxcorrection/limiting methods). In Section 2.2 we present the statement of the problemand the HRPG method in higher-dimensions. The statement in higher-dimensions ismade only to distinguish the current method with the existing ones. The structureof the method in 1d is identical to the CAU method except for the definitions ofthe stabilization parameters. The method can be derived via the FIC approach [141]with an adequate (nonlinear) definition of the characteristic length. Thus the resultspresented here may be extended to these methods. In Section 2.4 we focus on theGibbs phenomenon that is observed in L2 projections. The design procedure embarksby defining a model L2 projection problem and establishing the expression for thestabilization parameter to circumvent the Gibbs phenomenon. The target solution forthe model problem is chosen to be the one obtained via the mass-lumping procedure.We remark that this solution is used to evaluate the stabilization terms introduced bythe HRPG method and an expression for the stabilization parameter is defined thatdepends only on the problem data. In Section 2.5 we extend the methodology to thetransient convection–diffusion–reaction problem. We split the design into four modelproblems and derive the stabilization parameters accordingly. Finally we arrive at anexpression for the stabilization parameters depending only on the problem data andrepresenting asymptotically the prior expressions derived for the model problems.We summarize the HRPG design in Section 2.5.6. In Section 2.5.7 several examplesare presented that support the design objectives i. e.stabilization with high-resolution.Finally we arrive at some conclusions in Section 2.6.

2.2 high-resolution petrov–galerkin method

The statement of the multidimensional convection–diffusion–reaction problem is asfollows:

R(φ) :=∂φ

∂t+ u ·∇φ−∇ · (k∇φ) + sφ− f(x) = 0 in Ω (2.1a)

φ(x, t = 0) = φ0(x) in Ω (2.1b)

φ = φp on ΓD (2.1c)

(k∇φ) · n + gp = 0 on ΓN (2.1d)

where u is the convection velocity, k,s are the diffusion and reaction coefficient respec-tively, f(x) is the source, φ0(x) is the initial solution, φp and gp are the prescribedvalues of φ and the diffusive flux at the Dirichlet and Neumann boundaries respec-tively and n is the normal to the boundary.

The variational statement of the problem (2.1) can be expressed as follows: Findφ : [0, T ] 7→ V such that ∀w ∈ V0 we have,

(w,R(φ)

)Ω

+(w, (k∇φ) · n + gp

)ΓN

= 0 (2.2)


where, if H is the associated Hilbert space then V := w : w ∈ H and w = φp on ΓD,V0 := w : w ∈ H and w = 0 on ΓD, (·, ·)Ω and (·, ·)ΓN denote the L2(Ω) and L2(ΓN)inner products respectively. The problem (2.1) may also be expressed in the weakform as follows: Find φ : [0, T ] 7→ V such that ∀w ∈ V0 we have,

a(w,φ) = l(w) (2.3a)

a(w,φ) :=(w,∂φ

∂t+ u ·∇φ+ sφ

)Ω

+(∇w,k∇φ

)Ω

(2.3b)

l(w) :=(w, f(x)

)Ω

−(w,gp

)ΓN

(2.3c)

The statement of the Galerkin method applied to the weak form of the problem(2.3) is: Find φh : [0, T ] 7→ Vh such that ∀wh ∈ Vh0 we have,

a(wh,φh) = l(wh) (2.4)

We follow [92] to describe a certain class of Petrov–Galerkin methods which accountfor weights that are discontinuous across element boundaries. The perturbed weight-ing function is written as wh = wh + ph, where ph is the perturbation that accountfor the discontinuities. The statement of these class of Petrov–Galerkin methods is asfollows: Find φh : [0, T ] 7→ Vh such that ∀wh ∈ Vh0 we have,

a(wh,φh) +∑e

(ph,R(φh)

)Ωeh

= l(wh) (2.5)

The HRPG method whose design in 1d is presented in the subsequent sections, maybe defined as Eq.(2.5) along with the following definitions:

ph := [h + H · ur] ·∇wh (2.6a)

ur :=R(φh)

|∇φh|2∇φh; ⇒ ur :=

ur

|ur|=

sgn[R(φh)]|∇φh|

∇φh (2.6b)

where, h and H are frame-independent linear characteristic length tensors that are de-fined based on the element geometry (see Section 3.3). We refer to the Table(2) for acomparison of the HRPG method with the SUPG, FIC and some of the existing shock-capturing methods. From Eqs.(2.6),(2.7) and Table(2) the HRPG method could be un-derstood as the combination of upwinding plus a nonlinear discontinuity-capturingoperator. The distinction is that in general the upwinding provided by h is not stream-line and the discontinuity-capturing provided by H · ur is neither isotropic nor purelycrosswind. Of course defining h := τu and H := (β`)I or H := (β`)[I − u⊗ u] onewould recover (except for the definitions of the stabilization parameters) the CAU andthe CD methods respectively. We remark that one may arrive at the HRPG methodvia the finite-calculus (FIC) equations wherein the characteristic length is defined ashfic := h + H · ur. From this point of view the HRPG method can be presented as‘FIC-based’. More details are given in the next section.

Note that in 1d u‖ = u and hence the performance of the DC method [94] is similarto that of the SUPG method. Also note that as the notion of crosswind directions doesnot exist in 1d, the CD method [29] is identical to the SUPG method. On the otherhand the nonlinear shock-capturing terms introduced by the CAU method still existsin 1d and thus in principle are able to control the Gibbs and dispersive oscillations.


Method Perturbation (ph) Remarks

SUPG[92] τu ·∇wh

MH[132] CeiCei ∈ −

1

3,2

3,i = 1, 2, 3∑

Cei = 0

DC[94] τ1u ·∇wh + τ2u‖ ·∇wh u‖ :=u · ∇φh|∇φh|2

∇φh

CAU[68],τ1u ·∇wh + τ2ur ·∇wh ur :=

R(φh)

|∇φh|2∇φh

CCAU[50]

CD[29] τ1u ·∇wh +α2`∇wh · [I − u⊗ u] · ur u :=u|u|

ur :=ur

|ur|=

sgn[R(φh)]|∇φh|

∇φh

SAUPG[48],τ[λu + (1− λ)ur] ·∇wh λ is a smoothness measure.

Mod.CAU[29]

FIC[141] hfic ·∇wh here hfic is a characteristiclength vector which may bedefined in a linear or nonlin-ear fashion.

HRPG [h + H · ur] ·∇wh h,H are frame-independentlinear characteristic lengthtensors based on the elementgeometry (see Section 3.3).

Table 2: Perturbations associated with Petrov–Galerkin methods


This feature does carry over to all the methods that have the shock-capturing termsimilar to that in the CAU method viz. the methods presented in [23, 48–50]. Unfortu-nately as pointed out in [114] and in Section 2.5.7.1 of this chapter, these methods areoften over diffusive. The structure of the HRPG method in 1d is identical to the CAUmethod except for the definitions of the stabilization parameters. In the subsequentsections we design the stabilization parameters of the HRPG method to overcome theshortcomings of the earlier methods.

From Eqs.(2.5) and (2.6) the statement of the HRPG method in 1d can be expressedas follows: Find φh : [0, T ] 7→ Vh such that ∀wh ∈ Vh0 we have,

a(wh,φh)+∑e

[(α`2

dwhdx

,R(φh))Ωeh

+(β`2

|R(φh)|

|∇φh|dwhdx

,dφhdx

)Ωeh

]= l(wh) (2.7)

where α , β are stabilization parameters to be defined later.

2.3 derivation of the hrpg expression via the fic procedure

The governing equations in the finite calculus (FIC) approach are derived by express-ing the balance equations in a domain of finite size and retaining higher order terms.For the 1d convection–diffusion–reaction problem the FIC governing equations arewritten as [141]

R(φ) −h

2

∂R(φ)

∂x= 0 in Ω (2.8a)

φ(x, t = 0) = φ0(x) in Ω (2.8b)

φ−φp = 0 on ΓD (2.8c)

k∂φ

∂x+ gp +

h

2R(φ) = 0 on ΓN (2.8d)

where the characteristic length h is the dimension of the domain where balance offluxes is enforced and R(φ) is defined in Eq.(2.1a).

The variational statement of Eqs.(2.8) can be written as: Find φ : [0, T ] 7→ V suchthat ∀w ∈ V0 we have,

(w,R(φ) −

h

2

∂R(φ)

∂x

)Ω

+(w,k

∂φ

∂x+ gp +

h

2R(φ)

)ΓN

= 0 (2.9)

The corresponding weak form is: Find φ : [0, T ] 7→ V such that ∀w ∈ V0 we have,

(w+

h

2

dwdx

,R(φ))Ω

+(w,k

∂φ

∂x+ gp

)ΓN

= 0 (2.10)

In the derivation of Eq.(2.10) we have neglected the change of h within the elements.Clearly Eq.(2.10) can be seen as a Petrov–Galerkin form with the weighting functiondefined as w := w + h

2dwdx . The term depending on h in Eq.(2.10) is usually com-

puted in the element interiors only to avoid the discontinuities of the second deriva-tives terms in R(φ) along the element boundaries. The discretized form of Eq.(2.10) istherefore written as: Find φh : [0, T ] 7→ Vh such that ∀wh ∈ Vh0 we have,

a(wh,φh) +∑e

(h2

dwhdx

,R(φh))Ωeh

= l(wh) (2.11)


The characteristic length can be defined in a number of ways so as to provide an ‘op-timal’ (stabilized) solution. In this work the following nonlinear expression is chosenfor h:

h := α`+β`sgn[R(φh)]

|∇φh|∇φh (2.12)

where ` is the element length and α,β are stabilization parameters. A discussion ofthe alternatives for the definition of h in the FIC context can be found in [141, 152, 154,155]. Substituting the expression of h into Eq.(2.11) gives the HRPG form of Eq.(2.7).

2.4 gibbs phenomenon in L2 projections

2.4.1 Introduction

Gibbs phenomenon is a spurious oscillation that occurs when using a truncatedFourier series or other eigen function series at a simple discontinuity. It is character-ized by an initial overshoot and then a pattern of undershoot-overshoot oscillationsthat decrease in amplitude further from the discontinuity. In fact for any given func-tion f and using the metric as the standard L2 norm, the partial sum of order N of theFourier series of f denoted as SNf is the best approximation of f in a subspace spannedby trigonometric polynomials of order N. Thus SNf is the L2 projection of f in the con-sidered subspace. This phenomenon is manifested due to the lack of completeness ofthe approximation space. Similar oscillations appear in the problem of finding the bestapproximation of a given discontinuous function in any subspace using the L2 normas the metric. On every discrete grid/mesh the maximum wavenumber that can berepresented is limited by the Nyquist limit. The Nyquist frequency on a uniform gridwith grid spacing ` is given by π/`. Thus the span of the finite element basis functionsassociated with this mesh might be viewed as a truncated function series which mightbe expanded by refining the mesh. Hence the projection of a discontinuous functiononto this finite element space exhibits the Gibbs phenomenon. As the amplitude spec-trum of a discontinuous function decays only as fast as the harmonic series, which isnot absolutely convergent, it is impossible to circumvent these oscillations by meremesh refinement.

The variational statement of the L2 projection problem is as follows: Find φh ∈ Vhsuch that ∀wh ∈ Vh0 we have,

(wh,φh − f

)Ωh

= 0 (2.13)

Where f is the given function which might admit discontinuities. If we denote thesolution of Eq.(2.13) as φh = P0hf, we have,

‖ P0hf− f ‖L2(Ωh)6‖ wh − f ‖L2(Ωh) (2.14)

In the following sections we consider the scaled L2 projection problem defined bythe residual R(φ) = sφ− f. In the context of the convection–diffusion–reaction prob-lem, the problem data s physically represents the reaction coefficient. The variationalstatement of the scaled L2 projection problem is as follows: Find φh ∈ Vh such that∀wh ∈ Vh0 we have,

(wh, sφh − f

)Ωh

= 0 (2.15)


Taking s = 1 we recover the L2 projection problem given by Eq.(2.13).

2.4.2 Galerkin Method

2.4.2.1 FE discretization

Discretization of the space by linear finite elements will lead to the approximationφh = NaΦa and the Eq.(2.15) reduces into the following system of equations.

sM ·Φ = f (2.16)

Mab =(Na,Nb

)Ωh

; fa =(Na, f

)Ωh

(2.17)

It is well known that the Gibbs oscillations can be circumvented in the numericalsolution if the standard row-lumping technique is performed on the mass matrixM. Unfortunately, this operation though effective for this specific problem, it cannotbe extended to other problems in general. The answer for not advocating this tech-nique can be found in the Godunov’s theorem: ‘All linear monotone schemes are atmost first order accurate’. The only way to circumvent this problem is to design anonlinear method that would reproduce the same numerical solution as obtained bymass-lumping.

2.4.2.2 Model Problem 1

The 1d domain is chosen to be of unit length and discretized by 4N linear elements(N > 5). The function whose L2 projection is sought is defined as follows:

f(x) =

0 ∀ x ∈ [0, 0.25+ η1`]∪ [0.75− η2`, 1]q else

(2.18)

Where ` = 1/(4N) is the element length and η1,η2 ∈ [0, 1] are parameters that de-termine the location of the simple discontinuity in the function f. The solution ofEq.(2.16) using a lumped mass matrix can be expressed as follows:

Φ = (q

s)0, · · · , 0,

(1− η1)2

2,(2− η21)

2, 1, · · · , 1,

(2− η22)

2,(1− η2)

2

2, 0, · · · , 0 (2.19)

Figure (18a) illustrates the function f(x) using η1 = 0.5 , η2 = 0.3 and q = s =

1 alongside the numerical solution of the Eq.(2.16) using both the consistent andlumped mass matrix. Figure (18b) illustrates the profile characteristics of the mono-tone solution obtained via mass-lumping (Eq.2.19) with respect to the location of thediscontinuity.

2.4.2.3 Discrete Upwinding

Let the discrete system of equations be represented in the matrix form as follows:

A · x = b (2.20)

Discrete upwinding is an algebraic operation to convert the system matrix A into anM-matrix [120]. Discrete upwinding is the least diffusive linear operation to produce


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

x

φ hDesign Problem ; η

1 = 0.5, η

2 = 0.3, ∆ x = 0.025

GalerkinLumped Massf

(a) (b)

Figure 18: (a) The design problem ; (b) Characteristics of the monotone solution (AGFE). AB-HIJE illustrates the discontinuous regime of f(x)

an M-matrix. We denote the discrete upwinding operation on any given matrix A byDU(A). The discrete upwinding operation is performed by adding to the matrix A adiscrete diffusion matrix D as follows:

dii = −∑j6=i

dij; dij = dji = −max0,aij,aji (2.21)

DU(A) = A = A + D (2.22)

It is interesting to note that the discrete upwinding operation on the mass matrix Mwill result in the mass-lumping operation.

Me =`

6

[2 1

1 2

]; De =

`

6

[1 −1

−1 1

](2.23)

DU(Me) = Me + De =`

6

[3 0

0 3

]= Me

L (2.24)

2.4.2.4 Total variation

The total variation of a function, say φ(x), in 1d is given by the following equation:

TV(φ) =

∫x

|∇φ|dx (2.25)

Thus it can be seen that the total variation, as the name suggests, measures the totalhike or drop in the function profile as we traverse the 1d domain. It can also benoticed that any spurious oscillation in the numerical approximation of φ wouldcause the total variation to increase. Harten proved that a monotone scheme is totalvariation non-increasing (TVD) and a TVD scheme is monotonicity preserving [82]. Todate various high-resolution schemes have been designed based on the TVD concept


often using flux/slope limiters. If linear finite elements are used to approximate thenumerical solution φh the total variation may be calculated as follows:

TV(φh) =∑i

|Φi+1 −Φi| (2.26)

For the problem under consideration, the sufficient conditions given by Harten in[82] for a numerical scheme to be TVD drops down to the condition for the systemmatrix to be an M-matrix. Thus in the design of the high-resolution Petrov–Galerkinmethod we use the total variation of the numerical solution as a posteriori verificationcondition. The total variation of the given discontinuous function f(x) in the designproblem is TV(f) = 2.

2.4.3 HRPG design

In this section we design the stabilization parameters of the HRPG method given byEq.(2.7) and choosing α = 0. For the model problem described in Section 2.4.2.2 thestatement of the method is as follows: Find φh ∈ Vh such that ∀wh ∈ Vh0 we have,

(wh,R(φh)

)Ωh

+∑e

(β`2

|R(φh)|

|∇φh|dwhdx

,dφhdx

)Ωeh

= 0 (2.27)

Where, R(φh) := sφh − f is the residual and β is a stabilization parameter to bedefined later. If the domain is discretized by linear finite elements the Eq.(2.27) can beexpressed in the matrix form for each element as follows:

[sMe + Se

]·Φe = feg (2.28)

where the corresponding matrices are defined as,

Me =(Na,Nb

)Ωeh

=s`

6

[2 1

1 2

](2.29)

Se = (β`

2)( |R(φh)||∇φh|

dNa

dx,

dNb

dx

)Ωeh

=k∗(φh)`

[1 −1

−1 1

](2.30)

feg =(Na, f

)Ωeh

(2.31)

k∗(φh) =β

2

( |R(φh)||∇φh|

, 1)Ωeh

(2.32)

fg = q` 0, · · · , 0,(1− η1)

2

2,(2− η21)

2, 1, · · · , 1,

(2− η22)

2,(1− η2)

2

2, 0, · · · , 0 (2.33)

In order to design the parameter β we assume that the method converges to thesolution given by Eq.(2.19). This is a fair assumption as it can be seen in Eq.(2.27)that the nonlinear Petrov–Galerkin term is symmetric subjected to the linearizationas shown in Eq.(2.30) and hence there exists a β such that the effect of this term isequivalent to the discrete diffusion introduced by the discrete upwinding operation.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10.82

0.8333

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

g(η)

η

g(η) vs. η

Figure 19: The plot of g(η) for η ∈ [0, 1].

If η ∈ [0, 1] be a generic parameter to define the location of the simple discontinuitywithin the element, we have then for the element containing the discontinuity,

( |R(φh)||∇φh|

, 1)Ωeh

= (s`2

2)[1+ 2η− 6η2 + 8η3 − 4η4

1+ 2η− 2η2

]

= (s`2

2)[1+ 2η(1− η)[1− 2η(1− η)]

[1+ 2η(1− η)]

](2.34)

For the element adjacent to the element containing the discontinuity we have,

( |R(φh)||∇φh|

, 1)Ωeh

= (s`2

2) (2.35)

Thus, the nonlinear term in Eq.(2.27) and for the converged solution given byEq.(2.19) can be expressed as follows:

( |R(φh)||∇φh|

, 1)Ωeh

=

(s`2/2) g(η) , for elements with shock

(s`2/2) , for elements adjacent to the shock

0 , else

(2.36)

where, the function g(η) is defined as,

g(η) :=[1+ 2η(1− η)[1− 2η(1− η)]

[1+ 2η(1− η)]

](2.37)

∀ η ∈ [0, 1] , g(η) ∈ [(5/6), 1] (2.38)

Figure (19) illustrates the plot of the function g(η) vs η. To define the parameterβ we require that for the elements in the vicinity of the discontinuity the nonlinearPetrov–Galerkin method reproduces the effect of discrete upwinding. The system ma-trix for the element containing the discontinuity is as follows,

[sMe + Se

]= (

s`

6)

[2 1

1 2

]+ (βs`g(η)

4)

[1 −1

−1 1

](2.39)


To reproduce the effect of discrete upwinding the following relation should hold,

sMe + Se = DU(sMe) = sMe + De (2.40)

⇒ (βs`g(η)

4)

[1 −1

−1 1

]=

s`

6

[1 −1

−1 1

](2.41)

The expression for the parameter β may be expressed as follows:

βg(η) >2

3(2.42)

Remarks:

• The definition of β satisfying the equation βg(η) = 23 would exactly reproduce

the solution of Eq.(2.16) using the lumped mass matrix.

• From the design point-of-view the definition of β involving the function g(η)would imply a priori knowledge of the solution. Hence we define β using theextremum values of the function g(η).

Thus from Eq.(2.38) and Eq.(2.42) we have,

Type-I : ming(η) =5

6⇒ β >

4

5(2.43)

Type-II : maxg(η) = 1 ⇒ β >2

3(2.44)

With these definitions the design of the high-resolution Petrov–Galerkin (HRPG)method for the scaled L2 projection problem is complete.

2.4.4 Examples

2.4.4.1 Example 1

We solve the problem described in Section 2.4.2.2 and using q = s = 1. The 1d domainis discretized into 40 linear elements. The numerical results of the HRPG method(Type I and II) are compared with the solutions obtained by the Galerkin methodusing both consistent and lumped mass matrix. Figs.(20,21) illustrate the results ofHRPG Type I and Type II respectively for η1 = 0.5,η2 = 0.3. Figs.(22,23) illustrate thesame for η1 = 1,η2 = 0. Both Type I and Type II effectively circumvent the Gibbs phe-nomenon. HRPG Type I method is clearly more diffusive, nevertheless monotonicityis guaranteed. HRPG Type II is monotone to-the-eye. A quantitative analysis based onthe measured total variation is studied in Section 2.4.4.3.

2.4.4.2 Example 2

The analysis domain is the same as the problem described in Section 2.4.2.2 and usings = 1. The 1d domain is discretized into 100 linear elements. The function whose L2

projection is sought is now defined as follows:

f(x) =

cos(4πx− 2π) ∀ x ∈ [0.25, 0.75]

0 else(2.45)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

x

φ h

TYPE I: β = 4/5 ; η1 = 0.5, η

2 = 0.3

GalerkinLumped MassHRPG

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Err

or

Iteration

TYPE I: β = 4/5 ; η1 = 0.5, η

2 = 0.3

(b)

Figure 20: Example 1: HRPG Type I, (a) numerical solution for η1 = 0.5,η2 = 0.3 ; (b) corre-sponding nonlinear convergence plot

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

x

φ h

TYPE II: β = 2/3; η1 = 0.5, η

2 = 0.3


(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Err

or

Iteration

TYPE II: β = 2/3; η1 = 0.5, η

2 = 0.3

(b)

Figure 21: Example 1: HRPG Type II, (a) numerical solution for η1 = 0.5,η2 = 0.3 ; (b) corre-sponding nonlinear convergence plot


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

x

φ hTYPE I: β = 4/5 ; η

1 = 1.0, η

2 = 0.0


(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Err

orIteration

TYPE I: β = 4/5 ; η1 = 1.0, η

2 = 0.0

(b)

Figure 22: Example 1: HRPG Type I, (a) numerical solution for η1 = 1.0,η2 = 0.0 ; (b) corre-sponding nonlinear convergence plot

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

x

φ h

TYPE II: β = 2/3; η1 = 1.0, η

2 = 0.0


(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Err

or

Iteration

TYPE II: β = 2/3; η1 = 1.0, η

2 = 0.0

(b)

Figure 23: Example 1: HRPG Type II, (a) numerical solution for η1 = 1.0,η2 = 0.0 ; (b) corre-sponding nonlinear convergence plot


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.2

−1.1

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

x

φ hTYPE I: β = 4/5

GalerkinHRPG

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Err

orIteration

TYPE I: β = 4/5

(b)

Figure 24: Example 2: HRPG Type I, (a) numerical solution with both smooth and shockregimes ; (b) corresponding nonlinear convergence plot

The above function has both smooth and shock regimes. Simple discontinuities arepresent at x = 0.25 and x = 0.75 and in the rest of the domain the function is smooth.This example studies the efficiency of the HRPG method for mixed regimes. Figures(24,25) illustrate that the accuracy of the solution in the smooth regime is not compro-mised while effectively circumventing the Gibbs phenomenon around the shocks.

2.4.4.3 Example 3

The problem considered in this example is the same as in Section 2.4.4.1. To reduce thevariability of the problem data, we have chosen η1 = η2. As it is mentioned earlier inSection 2.4.2.4, the total variation (TV) of the numerical solution (φh) is a direct mea-sure (though a posteriori) of the presence of spurious oscillations. HRPG Type I methodguarantees that the system matrix for the current problem is an M-matrix for all valuesof η1 and η2. This is a sufficient condition to obtain a monotonicity-preserving solu-tion. The system matrix using the HRPG Type II method is an M-matrix only whenη1,η2 = 0, 1. Thus we study TV(φh) to have a quantitative measure of performancefor the Type I and Type II methods.

Figures (26,27) illustrate with respect to the Galerkin method the TV(φh) vs η plotsfor the HRPG Type I and Type II methods respectively. It is remarkable that boththe methods measure TV(φh) = 2 which is the same as TV(f). A study of the errorTV(φh) − TV(f) suggests (as expected) that for the Type I method TV(φh) < TV(f)

and TV(φh) − TV(f) = O(1e-11). For the Type II method, TV(φh) > TV(f) andTV(φh) − TV(f) = O(1e-5) which is an acceptable tolerance. In the light of theseresults the method we currently prefer is HRPG Type II.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.2

−1.1

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

x

φ hTYPE II: β = 2/3

GalerkinHRPG

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Err

orIteration

TYPE II: β = 2/3

(b)

Figure 25: Example 2: HRPG Type II, (a) numerical solution with both smooth and shockregimes ; (b) corresponding nonlinear convergence plot

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.5

1

1.5

2

2.5

3

3.5

4

η

TV

(φh)

TYPE I: β = 4/5TV(φ

h),Galerkin

TV(φh),HRPG

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1−10

−11

−10−12

−10−13

η

TV

(φh)−

TV

(f)

HRPG TYPE I: β = 4/5TV(φ

h)−TV(f)

(b)

Figure 26: Example 3: HRPG Type I, (a) TV(φh) plot ; (b) TV(φh)-TV(f) plot


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.5

1

1.5

2

2.5

3

3.5

4

η

TV

(φh)

TYPE II: β = 2/3TV(φ

h),Galerkin

TV(φh),HRPG

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 110

−9

10−8

10−7

10−6

10−5

10−4

ηT

V(φ

h)−T

V(f

)

HRPG TYPE II: β = 2/3TV(φ

h)−TV(f)

(b)

Figure 27: Example 3: HRPG Type II, (a) TV(φh) plot ; (b) TV(φh)-TV(f) plot

2.4.5 Summary

Residual : R(φh) := sφh − f

HRPG method :(wh,R(φh)

)Ωh

+∑e

(β`2

|R(φh)|

|∇φh|dwhdx

,dφhdx

)Ωeh

= 0

Type I : β >4

5→M-matrix guaranteed

Type II : β >2

3→ Total variation limit

The Gibbs phenomenon that arises in L2 projections is studied for the Galerkinmethod in 1d using linear finite elements. A nonlinear Petrov–Galerkin method (HRPG)is formulated and the stabilization parameter is designed (Type I and Type II) so asto circumvent the Gibbs phenomenon and thus leading to a high-resolution method.The HRPG method is shown to perform well in the presence of both smooth andshock regimes in the solution. HRPG Type II method is shown to be in the total varia-tion limit with acceptable tolerance of O(1e-5) and thus is essentially non-oscillatory(monotone to-the-eye). Hence it is currently the preferred choice for the extension ofthe HRPG design to the model problems in the subsequent sections.

2.5 convection–diffusion–reaction problem

2.5.1 Galerkin method and discrete upwinding

Consider the convection–diffusion–reaction problem given by Eq.(2.1) in 1d and sub-jected only to the Dirichlet boundary conditions. Discretization of the space by linear


finite elements will lead to the approximation φh = NaΦa. For the Galerkin methodwe arrive at the following system of equations.

MΦ+ [uC + kD + sM]Φ = f (2.46)

where the element contributions to the above matrices and vector are given by,

Meab =

(Na,Nb

)Ωh

=`

6

[2 1

1 2

], Ceab =

(Na,∇(Nb)

)Ωh

=1

2

[−1 1

−1 1

]

(2.47)

fea =(Na, f

)Ωh

=`

2

1

1

, Deab =

(∇(Na),∇(Nb)

)Ωh

=1

l

[1 −1

−1 1

]

(2.48)

The discrete upwinding process applied to the steady state of Eq.(2.46) will intro-duce a numerical diffusion kdu as follows:

DU(uC + kD + sR) = [uC + kD + sR] + kduD (2.49)

kdu = max[

|u|`

2+s`2

6− k

], 0

= k max[

|γ|+ω

6− 1]

, 0

(2.50)

The form of this numerical diffusion (Eq.(2.50)) is identical to that found in [152] usinga FIC-based approach. The stabilization method presented in [152] introduces withineach element an additional nonlinear diffusion as follows:

kfic = k max[(

sgn[∇φh]sgn[∆φh]

)γ+

(sgn[φh]

sgn[∆φh]

)ω

6− 1

], 0

(2.51)

Clearly the form of Eq.(2.50) is an upper bound of the value of kfic as defined inEq.(2.51).

2.5.2 Model problem 2

Consider the steady diffusion–reaction problem with a distributed source term givenby Eq.(2.18) and homogeneous Dirichlet boundary conditions:

R(φ) := −k∆(φ) + sφ− f(x) (2.52)

For the current problem we design the HRPG method with α = 0. In the limit ask→ 0 the problem reduces to the scaled L2 projection problem considered in Section2.4. The discrete upwinding operation on the Galerkin method, will introduce anartificial diffusion equivalent to the following,

kdu = max[s`2

6− k

], 0

(2.53)

Note that ∀k 6 (s`2/6) the critical non-oscillatory solution obtained via the discreteupwinding process is identical to the solution obtained with k = 0. Thus in orderto design the parameter β we can use the solution given by Eq.(2.19) to estimate the


amount of nonlinear diffusion that would be introduced by the HRPG method. Thus,

k∗(φh) :=β

2

( |R(φh)||∇φh|

, 1)Ωeh

= βs`2

4g(η) > max

[s`2

6− k

], 0

(2.54)

We define a dimensionless element number ω := (s`2/k) and consider g(η) = 1

(Type-II). Thus,

β > max2

3

[1−

6

ω

], 0

(2.55)

We remark that β depends only on the problem data and for the current modelproblem the nonlinear (residual-based) diffusion k∗(φh) implemented in the HRPGmethod is nonzero and equals kdu only for the elements in the vicinity of the discon-tinuity. In this way it differs from the form of Eq.(2.51) which for the current modelproblem introduces a nonlinear diffusion kfic (Eq.(2.51) using γ = 0) for all elements.


Consider the steady convection-diffusion problem,

R(φ) := u∇(φ) − k∆(φ) (2.56)

For the current problem we design the HRPG method with α = 0. The discrete up-winding operation on the Galerkin method, will introduce an artificial diffusion equiv-alent to the following,

kdu = max[

|u|`

2− k

], 0

(2.57)

The parameter β may be designed as follows:

|R(φh)|

|∇(φh)|=

|u∇(φh)||∇(φh)|

= |u| (2.58)

⇒ k∗(φh) :=β

2

( |R(φh)||∇(φh)|

, 1)Ωeh

=β

2|u|` > max

[|u|`

2− k

], 0

(2.59)

⇒ β > max[1−

1

|γ|

], 0

(2.60)

Note that in contrast to the problem considered in Section 2.5.2 here we do not needthe solution to estimate the nonlinear diffusion introduced by the HRPG method. Theexpression for β (Eq.2.60) is identical to the standard critical stabilization parameterobtained for upwind techniques (see [195]).


Consider the steady convection–diffusion–reaction problem:

R(φ) := u∇(φ) − k∆(φ) + sφ− f(x) (2.61)


For the current problem we design the HRPG method again with α = 0. If linear finiteelements were used, the residual obeys the following relation:

|R(φh)| = |u∇(φh) + sφh − f| 6 |u∇(φh)|+ |sφh − f| (2.62)

⇒ |R(φh)|

|∇(φh)|=

∣∣∣∣u+sφh − f

∇(φh)

∣∣∣∣ 6 |u|+|sφh − f|

|∇(φh)|(2.63)

As it can be seen in Eq.(2.63), to estimate the nonlinear diffusion introduced by theHRPG method we require the solution of the problem a priori. The simplest ideawould be to use the nodally exact solution. Unlike the solution used in Section 2.5.2,the analytical solution of the current problem has a complex structure [152]. In orderto retain simplicity in the design of β we make a conjuncture of the results obtainedin Section 2.5.2 and Section 2.5.3. The conjuncture is made such that the designedexpression for β would approach asymptotically the expressions obtained in Section2.5.2 and Section 2.5.3 as u→ 0 and s→ 0 respectively.

Assume that u s and f(x) be defined as in Eq.(2.18). Thus we may approximatethe solution of the current problem to the one considered in Section 2.5.2. This assump-tion allows us to use the solution defined by Eq.(2.19) to approximately estimate thefollowing expression:

( |sφh − f|

|∇(φh)|, 1)Ωeh

≈ s`2

2g(η) (2.64)

As u s we make another approximation using Eq.(2.63) as follows:

|R(φh)|

|∇(φh)|≈ |u|+

|sφh − f|

|∇(φh)|(2.65)

Using the two approximations (Eq.2.64, Eq.2.65) the parameter β may be designed asfollows:

k∗(φh) :=β

2

( |R(φh)||∇(φh)|

, 1)Ωeh

≈ β2

[|u|`+

s`2

2g(η)

]> max

[|u|`

2+s`2

6− k

], 0

(2.66)

β := max[2

3

(|σ|+ 3

|σ|+ 2

)−

(4

ω+ 4|γ|

)], 0⇒

limu→0

β = max2

3

[1−

6

ω

], 0

lims→0

β = max[1−

1

|γ|

], 0

(2.67)

where, γ := (u`/2k) , ω := (s`2/k) and σ := (ω/2γ) = (s`/u) are the element Peclectnumber, a velocity independent dimensionless number and the Damköler numberrespectively.Remark : Eq.(2.66) does not mean that k∗(φh) = kdu for all elements and problemdata. We remind that under the assumption u s and f(x) defined as in Eq.(2.18), thesolution to the current model problem is approximated as the one given by Eq.(2.19).This suggests that, similar to the model problems in Section 2.4.2.2 and Section 2.5.2,the nonlinear (residual-based) diffusion k∗(φh) equals kdu only for the elements inthe vicinity of the layers. In general, as β is independent of φh, the only informationknown a priori is that k∗(φh) is proportional to the residual R(φh). The argument thatthis expression for β i. e.Eq.(2.67), would perform well ∀u, s is a conjecture based on


the fact that we recover asymptotically the expressions for β i. e.Eq.(2.55) and Eq.(2.60)as u → 0 and s → 0 respectively. The efficiency of this expression for β is shown viavarious numerical examples (see Section 2.5.7).


Consider the transient convection–diffusion–reaction problem:

R(φ) := φ+ u∇(φ) − k∆(φ) + sφ− f(x) (2.68)

We now design the parameter β associated with the nonlinear perturbation term whenthe linear perturbation terms exists (i. e.α 6= 0). The HRPG method after discretizationby linear finite elements will lead to the following system of equations.

[M]Φ+ [uC + kD + sM]Φ− fg (Galerkin terms)

+α`

2

([Ct]Φ+ [uD + sCt]Φ− fs

)(linear PG terms)

+β

2

( |R(φh)||∇(φ)| , 1

)Ωeh

[D]Φ = 0 (nonlinear PG terms) (2.69)

The above equation may be rearranged as follows

MΦ+ sΦ+CuΦ+Ctα`

2Φ+

α`s

2Φ+(k+

α`u

2+k∗(φh))DΦ = fg +

α`

2fs (2.70)

where the expression for k∗(φh) is given by Eq.(2.32). We define for each element ameasure δ with the dimensions of the reaction coefficient.

δ := ΦΦ ⇒ Φ = δΦ (2.71)

where the vector operators and are understood to operate point-to-point divi-sion and multiplication respectively. Thus in the design of the parameter β, we maymodel δ as a non-linear reactive coefficient. For the fully discrete problem (after timediscretization)δ may be approximated to an element-wise positive constant for simpli-fication. This idea had been pointed out earlier in [99]. Also note that the convectionmatrix C is skew-symmetric. Hence the transposed matrix Ct introduces a negativeconvection effect [30]. Following this line, for each element we define the effectiveconvection, diffusion and reaction coefficients as follows:

u := u−α`s

2−α`δ

2; k := k+

α`u

2; s := s+ δ (2.72)

The effective coefficients u, k and s will be used to design the parameter β. The fol-lowing effective element dimensionless numbers may be defined:

γ :=u`

2k; ω :=

s`2

k; σ :=

s`

u=ω

2γ(2.73)

The parameter β is now defined as in Eq.(2.67) using these effective element numbersas follows:

β := max[2

3

(|σ|+ 3

|σ|+ 2

)−

(4

ω+ 4|γ|

)], 0

(2.74)


If the discretization in time is done using the implicit trapezoidal rule, we have

Φ =Φ−Φn

θ∆t; Φn+1 =

1

θΦ+

θ− 1

θΦn (2.75)

where, ∆t is the time increment, n,n+ 1 denote the previous and current time stepsand θ ∈ [0, 1] is a parameter that defines the scheme. θ = 0, 0.5, 1 define the forwardEuler, implicit midpoint and the backward Euler methods. For the fully discrete sys-tem and within each element we evaluate the parameter δ and the residual as follows:

δ ≈ 1

θ∆t

‖ φh −φnh ‖e∞‖ φh ‖e∞ (2.76)

R(φh) ≈φh −φnhθ∆t

+ u∇(φh) − k∆(φh) + sφh − f (2.77)

where ‖ · ‖e∞ is the L∞norm. Note that as steady state is reached δ → 0. Thus for thesteady state problem and using α = 0 we recover the definition of the parameter β asgiven by Eq.(2.67).

It remains to define the parameter α that controls the fraction of linear perturbationterm in the HRPG method. For the 1d convection–diffusion–reaction problem, theHRPG method using α = 0 does solve a plethora of examples to give high-resolutionstabilized results. Nevertheless for the transient problem the presence of the linearperturbation terms improves the convergence of the nonlinear iterations. Numericalexperiments suggest α ∈ [0, 1/3] which means that the approximations/conjectureused in the design strategy does not hold for larger fractions of the linear perturbationterm. The following expression for α was used in the examples to come.

α := λ sgn(u)max[1−

1

|γ|

], 0

; λ :=1

3(1+√

|σ|)(2.78)


2.5.6 Summary

residual

R(φh) :=∂φh∂t

+ u∇(φh) − k∆(φh) + sφh − f(x)

the hrpg method

Find φh : [0, T ] 7→ Vh such that ∀wh ∈ Vh0 we have,

a(wh,φh)+∑e

(α`2

dwhdx

,R(φh))Ωeh

+(β`2

|R(φh)|

|∇φh|dwhdx

,dφhdx

)Ωeh

= l(wh)

PG weight→ wh +

[α`

2+β`

2sgn[R(φh)] sgn[∇(φh)]

]dwhdx

definitions

γ :=u`

2k; ω :=

s`2

k; σ :=

s`

u


+ u∇(φh) − k∆(φh) + sφh − f

φn+1n =(1θ

)φh +

(θ− 1θ

)φnh ; ∆t = tn+1 − tn ; θ ∈ (0, 1)

λ :=1

3(1+√

|σ|); δ ≈ 1

θ∆t

‖ φh −φnh ‖∞‖ φh ‖∞

α := λ sgn(u)max[1−

1

|γ|

], 0

u := u−α`s

2−α`δ

2; k := k+

α`u

2; s := s+ |δ|

γ :=u`

2k; ω :=

s`2

k; σ :=

s`

u

β := max[2

3

(|σ|+ 3

|σ|+ 2

)−

(4

ω+ 4|γ|

)], 0

2.5.7 Examples

2.5.7.1 Example 1

We consider the convection–diffusion–reaction problem given by Eq.(2.1) in 1d. Westudy the steady-state case with the following data : k = 1 and u, s 6= 0. The 1d

domain is taken as x ∈ [0, 1] and it is discretized with eight two-node linear elements.The values of u and s are determined appropriately for different values of γ and ω.The results of the HRPG method (using both λ = 0 and λ 6= 0) are compared with thatof the Galerkin, Galerkin with discrete upwinding (DU), SUPG, CAU, modified CAUand the FIC based stabilization method presented in [152]. The error in the nonlineariterations was measured by the following norm:

‖ Φi+1 −Φi ‖e

‖ Φi+1 ‖e(2.79)

where, ‖ · ‖e is the standard Euclidean vector norm. A tolerance of 1e-5 was chosenas the termination criteria. A maximum of 30 iterations were allowed. Note that thenumber of iterations required by the nonlinear methods for convergence is displayed


next to the corresponding legends. The nonlinear iterations were initialized by thesolution obtained by the DU method.

Figures 28 - 33 illustrate the solution obtained for the sourceless case (f = 0) and for(γ,ω) = (1, 5), (1, 20), (1, 120), (2, 2), (10, 4), (10, 20) respectively. The Dirichlet bound-ary conditions φpL := φ(x = 0) = 8 and φpR := φ(x = 1) = 3 were employed. The DUmethod is robust and provides stable solutions. Unfortunately the accuracy achievedis at most first-order and hence the solutions are generally over-diffusive. The FICmethod presented in [152] provides more accurate solutions and remarkably the non-linear iterations converge with just two iterations. Slight node-to-node oscillationsaround the exact solutions are observed for the case γ = 10, ω = 4 viz. Figure 32aand is duly discussed in [152]. As expected the SUPG method provides good solu-tions to all except the reaction-dominated cases viz. Figures 29b,30b. The CAU andmodified CAU methods succeed in circumventing the instabilities for the reaction-dominated cases but for these cases provide solutions that are more diffusive thanthat of the DU method. The HRPG method provides good solutions for all the casesconsidered. Note that for the reaction-dominated cases the solutions are less diffusivethan the DU, CAU and modified CAU methods. Also note that the solutions obtainedby taking λ = 0 is indistinguishable to that obtained by taking λ 6= 0. Nevertheless thenonlinear iterations converge faster for the latter.

Figures 34,35 illustrate the solution obtained for the sourceless case (f = 0) with(γ,ω) = (10, 200) and for Dirichlet boundary conditions (φpL,φpR) = (0, 1), (1, 0) re-spectively. The FIC method of [152] provides nodally exact to-the-eye solutions andthe nonlinear iterations converge within 2 iterations. The solutions obtained by theSUPG and CAU methods are indistinguishable and exhibit instabilities for the latterboundary conditions viz. Figure 35b. The modified CAU method circumvents these in-stabilities but instead provides solutions that are more diffusive than the DU method.The HRPG method (both λ = 0 and otherwise) succeed to provide stable solutionsand are less diffusive than that obtained by the DU method. Figure 34 shows that theHRPG method with λ = 0 converges in just one iteration while using λ 6= 0 seveniterations were needed. This is a rare coincidence where the initial solution providedby the DU method and the solution of the HRPG method with λ = 0 are closer thanthe specified tolerance.

Figures 36,37 illustrate the solution obtained with (γ,ω) = (2, 0) and (f,φpL,φpR) =(0, 0, 1), (u, 0, 0) respectively. As expected the SUPG method provides nodally exactsolutions for these cases. The DU, CAU and HRPG methods provide stable solutionsand are indistinguishable from each other. The modified CAU solution is very similarto the solutions of the former methods. For the sourceless case (f = 0) the solutionsof the DU and FIC methods are indistinguishable. Unfortunately when f 6= 0 thenonlinear iterations associated with the latter fail to converge. We believe that thisbehavior is due to the increased nonlinearity associated with the definition of thestabilization parameters (See [152]).

2.5.7.2 Example 2

We consider again the convection–diffusion–reaction problem given by Eq.(2.1) in 1d.Now we study the transient pure-convection problem, i. e.k, s, f = 0. The Dirichlet


0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−1

0

1

2

3

4

5

6

7

8

9

x

Φ

γ = 1, ω = 5AnalyticalGalerkinFIC; 2 Iter.HRPG(λ = 0); 5 Iter.HRPG(λ ≠ 0); 5 Iter.

(a)

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−1

0

1

2

3

4

5

6

7

8

9

x

Φ

γ = 1, ω = 5AnalyticalDUSUPGCAU; 18 Iter.Mod.CAU; 8 Iter.

(b)

Figure 28: Steady state: (γ,ω, f,φpL ,φpR) = (1, 5, 0, 8, 3). (a) Exact, Galerkin, FIC [152], HRPG(λ = 0) and HRPG (λ 6= 0) solutions; (b) Exact, DU, SUPG, CAU and Mod.CAUsolutions

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−2

−1

0

1

2

3

4

5

6

7

8

9

x

Φ


(a)

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−2

−1

0

1

2

3

4

5

6

7

8

9

x

Φ


(b)



0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

9

x

Φ


(a)

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

9

x

Φ


(b)


0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−1

0

1

2

3

4

5

6

7

8

9

x

Φ


(a)

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−1

0

1

2

3

4

5

6

7

8

9

x

Φ


(b)



0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 10

1

2

3

4

5

6

7

8

9

x

Φ


(a)

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 10

1

2

3

4

5

6

7

8

9

x

Φ


(b)


0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−3

−2

−1

0

1

2

3

4

5

6

7

8

9

x

Φ


(a)

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−3

−2

−1

0

1

2

3

4

5

6

7

8

9

x

Φ


(b)



0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

Φγ = 10, ω = 200

AnalyticalGalerkinFIC; 1 Iter.HRPG(λ = 0); 1 Iter.HRPG(λ ≠ 0); 7 Iter.

(a)

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

Φ


(b)

Figure 34: Steady state: (γ,ω, f,φpL ,φpR) = (10, 200, 0, 0, 1). (a) Exact, Galerkin, FIC [152],HRPG (λ = 0) and HRPG (λ 6= 0) solutions; (b) Exact, DU, SUPG, CAU andMod.CAU solutions

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

Φ


(a)

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

Φ


(b)

Figure 35: Steady state: (γ,ω, f,φpL ,φpR) = (10, 200, 0, 1, 0). (a) Exact, Galerkin, FIC [152],HRPG (λ = 0) and HRPG (λ 6= 0) solutions; (b) Exact, DU, SUPG, CAU andMod.CAU solutions


0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

Φ


(a)

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

Φ


(b)


0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

x

Φ

γ = 2, ω = 0, f = uAnalyticalGalerkinHRPG(λ = 0); 1 Iter.HRPG(λ ≠ 0); 1 Iter.

(a)

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

x

Φ

γ = 2, ω = 0, f = uAnalyticalDUSUPGCAU; 1 Iter.Mod.CAU; 4 Iter.

(b)

Figure 37: Steady state: (γ,ω, f,φpL ,φpR) = (2, 0,u, 0, 0). (a) Exact, Galerkin, HRPG (λ = 0) andHRPG (λ 6= 0) solutions; (b) Exact, DU, SUPG, CAU and Mod.CAU solutions


boundary condition φ(x = 0) = 0 is employed. The following initial condition wasused:

φ(x, t = 0) =

1 ∀ x ∈ [0.1, 0.2]∪ [0.3, 0.4]0 else

(2.80)

The above initial condition models a double rectangular pulse with simple discontinu-ities. The amplitude spectrum of this function decays only as fast as the harmonic seriesand hence is rich in high wave numbers. It is a challenging problem for the validationof any method for the control of dispersive oscillations and accuracy. The 1d domainis taken as x ∈ [0, 1] and it is discretized with 200 two-node linear elements. The timestep was chosen as ∆t = 0.001s. This corresponds to a Courant number C = 0.2. Theerror was measured using Eq.(2.79) and a tolerance of 1e-4 was used. For the HRPGmethod with λ = 0 a tolerance of 1e-3 was used. The nonlinear iterations at everytime step were initialized by the solution obtained by the SUPG method.

Figures 38,39 illustrate the solution obtained with the SUPG, CAU, modified CAUand HRPG methods. As expected the SUPG solution exhibits dispersive oscillationsviz. Figure 38a. Appreciable control over the dispersive oscillations is obtained inthe CAU method viz. Figure 38b. However slight crests and troughs do appear inthe solution that gradually die out in time. These crests and troughs are reducedin the solution obtained by the modified CAU method at the cost of accuracy viz.Figure 38c. Best results were obtained with the HRPG method with λ 6= 0 whichexhibits better control over the dispersive oscillations and maintains the symmetryof the initial solution viz. Figure 39b-c. On the other hand the HRPG method withλ = 0 is more diffusive than with λ 6= 0 and needs more iterations per time step forconvergence viz. Figure 39a.

2.6 conclusions

A high-resolution Petrov–Galerkin method is presented for the 1d convection–diffusion–reaction problem. The prefix ‘high-resolution’ is used here in the sense popularizedby Harten, i. e.second order accuracy for smooth/regular regimes and good shock-capturing in nonregular regimes. The HRPG method could be understood as thecombination of upwinding plus a nonlinear discontinuity-capturing operator. Thedistinction is that in general (multidimensions) the upwinding provided by h is notstreamline and the discontinuity-capturing provided by H · ur is neither isotropic norpurely crosswind. The HRPG form can be considered as a particular class of the sta-bilized governing equations obtained via a finite calculus (FIC) procedure. For the 1d

problem the HRPG method is similar to the CAU method with new definitions of thestabilization parameters. The 1d examples presented demonstrate that the methodprovides stabilized and essentially non-oscillatory i. e.monotone to-the-eye solutionsfor a wide range of the physical parameters and boundary conditions. It is interestingto note that the HRPG method without the linear upwinding term, i. e.using α = 0

does solve all the steady-state examples to give high-resolution stabilized results. Nev-ertheless the presence of the linear perturbation terms improves the convergence ofthe nonlinear iterations especially for the transient problem.

2.6 Conclusions 67

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Space

Φ

SUPG, Time = 0s, 0.1, 0.5s

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Space

Φ

CAU, Time = 0s, 0.1, 0.5s; SUPG Initialization

(b)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Space

Φ

Mod.CAU, Time = 0s, 0.1, 0.5s; SUPG Initialization

(c)

Figure 38: Transient pure convection; u = 1m/s, ` = 0.005m, ∆t = 0.001s. (a) SUPG solution;(b) CAU solution; (c) Mod.CAU solution


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Space

ΦHRPG (λ = 0), Time = 0s, 0.1, 0.5s; SUPG Initialization

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Space

Φ

HRPG (λ ≠ 0), Time = 0s, 0.1, 0.5s; SUPG Initialization

(b)

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

00.05

0.10.15

0.20.25

0.30.35

0.40.45

0.5

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time

HRPG (λ ≠ 0), Time = 0−0.5 s; SUPG Initialization

Space

Φ

(c)

Figure 39: Transient pure convection; u = 1m/s, ` = 0.005m, ∆t = 0.001s. (a) HRPG (λ = 0)solution; (b) HRPG (λ 6= 0) solution; (c) HRPG (λ 6= 0) solution (evolution plot)

yad bhavam tad bhavati.(As is the feeling, so is the result).

— a Sanskrit saying.

3M U LT I D I M E N S I O N A L E X T E N S I O N O F T H E H R P G M E T H O D

3.1 introduction

This chapter is a continuation of chapter §2 wherein a nonlinear high-resolutionPetrov–Galerkin (HRPG) method was presented for the convection–diffusion–reactionproblem in 1d. In this chapter we develop the extension to multi dimensions of theHRPG method for the singularly perturbed convection–diffusion–reaction problem.

It is well known that the solution to the stationary convection–diffusion–reactionproblem may develop two types of layers: exponential and parabolic layers. The expo-nential layers are usually found in the convection-dominant cases near the boundaryor close to the regions where the source term is non regular. Parabolic layers, whichare of larger width than exponential layers, are found in the reaction-dominant casesnear the boundary or close to the regions where the source term is non regular andin the convection-dominated cases along the characteristics of the solution. The latercharacteristic internal/boundary layers are usually found only in higher dimensionsand hence have no instances in 1d [175]. In other words we do not have a straight-forward quantification of the characteristic layers in 1d. For this reason a direct exten-sion of the definition of the stabilization parameters α,β derived for 1d will not beefficient to resolve these layers.

The numerical artifacts that are formed across the parabolic layers are usually man-ifested as the Gibbs phenomenon. Nevertheless there exists a subtle difference1 be-tween the numerical artifacts formed across the characteristic layers and those formedacross the layers in the reaction-dominant cases. Consider a rectangular domain dis-cretized by structured bilinear block finite elements. We use the following notation(introduced in [137]) to represent a generic compact stencil corresponding to any in-terior node (i,j) of the considered structured mesh.

j+1, j, j−1Ai−1, i, i+1t = 0 (3.1)

where A represents the matrix of the stencil coefficients. For instance, if the standardmass matrix obtained in the Galerkin FEM be assembled for a structured rectangularmesh then we may express the corresponding stencil as follows:

Am :=`261, 4, 1t

`161, 4, 1 =

`1`236

1 4 1

4 16 4

1 4 1

(3.2)

1 related to the cause and size of these numerical artifacts

69

70 3 multidimensional extension of the hrpg method

j+1, j, j−1Ami−1, i, i+1t := `1`236

Φi−1,j+1 + 4Φi,j+1 +Φi+1,j+1+

4Φi−1,j + 16Φi,j + 4Φi+1,j+

Φi−1,j−1 + 4Φi,j−1 +Φi+1,j−1

(3.3)

The stencil coefficient matrix associated with the convective term in the GalerkinFEM can be expressed as follows:

Ac :=`261, 4, 1t

u12

−1, 0, 1+u221, 0,−1t

`161, 4, 1 (3.4)

Note that one may arrive at the terms in Eq.(3.2) and Eq.(3.4) via a 1d mass type av-eraging of their respective counterparts in 1d, i. e.replacing (`1/6)1, 4, 1 with (`2/6)

1, 4, 1t (`1/6)1, 4, 1 and (u1/2)−1, 0, 1 with (`2/6)1, 4, 1t(u1/2)−1, 0, 1 etc. Al-though this 1d mass type averaging leads to a higher-order approximation for smoothsolution profiles, it unfortunately leads to the Gibbs phenomenon across layers. Un-like in the reaction-dominant case where it is the numerical solution that undergoesthe 1d mass type averaging, in the convection-dominant case it is the derivatives ofthe numerical solution that undergoes the same. Thus, the Gibbs phenomenon acrossthe characteristic layers in the later case is proportional to the variation in the deriva-tives of the solution across the characteristic layers. Despite this subtle difference inthe Gibbs phenomenon associated with the characteristic layers in the convection-dominated case, we choose to treat them by the same strategy that we use to treat thenumerical artifacts about the parabolic layers in the reaction-dominant case.

The outline of this chapter is as follows. In §3.2 we design a nondimensional ele-ment number that quantifies the characteristic internal/boundary layers. Anisotropicelement length vectors li are introduced in §3.3 and using them objective characteris-tic tensors h and H associated with the HRPG method are defined. The stabilizationparameters αi,βi used in the definition of h, H are defined in §3.4 by a direct exten-sion of their respective expressions in 1d. The definitions of βi are updated to includethe new dimensionless number introduced in §3.2. In Box 1 we summarize the HRPGmethod in multi dimensions. Several numerical examples are presented in §3.5 thatthrows light on the performance of the proposed method. Finally we arrive at someconclusions and outlook in §3.6

3.2 quantifying characteristic layers

In this section we design a nondimensional element number that quantifies the char-acteristic internal/boundary layers. By quantification we mean that its should serve asimilar purpose as the element Peclet number γ for the exponential layers in convec-tion dominant cases and the dimensionless number ω := 2γσ for the parabolic layersin the reaction dominant cases.

Consider the following singularly perturbed (k u) convection–diffusion problemin 2d:

u∂φ

∂x− k

(∂2φ

∂x2+∂2φ

∂y2

)= 0 in Ω (3.5a)

φ = φp on Γ (3.5b)


where, Ω is a rectangular domain ABCD as shown in Figure 40a, Γ is the domainboundary and φp is the prescribed value of φ on Γ . The origin of the 2d axes is takenas the midpoint of AD. Consider φp = 0 everywhere except on ADwhere it is definedas follows:

φp(0,y) = f(y) := H(y+ a) − H(y− a),a > 0 (3.6a)

H(y) :=1+ sgn(y)

2=

0 y < 0

0.5 y = 0

1 y > 0

(3.6b)

The function f(y) is discontinuous at y = ±a and its shape can be described as arectangular pulse. A well known virtue of the solution φ(x,y) is that these disconti-nuities are immediately smoothed out in the interior of the domain, thus leading toparabolic layers along the characteristic lines of the problem [175]. In accordance withsingular perturbation theory and by the method of matched asymptotic expansions[117], the leading term describing the characteristic layer is given by,

φ(x,y) ≈ 12

[erf(√

u

4kx(y+ a)

)− erf

(√u

4kx(y− a)

)](3.7)

where erf represents the error function and is defined as follows:

erf(x) :=2√π

∫x0

e−z2

dz (3.8)

The approximation given in Eq.(3.7) is uniformly valid to O(1) in a region awayfrom the exponential layers formed near the boundary BC [117]. Figure 40b illustratesthe solution given by Eq.(3.7) about a cross-section SS ′ (cf. Figure 40a) located at adistance x from the boundary AD.

A B

CD

u

x S

S’

φ(x,y) onboundary AD

(a)

C

D

B

A

x

u

φ(x,y) on boundary AD

φ(x,y) at an interiorcross−section SS’

(b)

Figure 40: A singularly perturbed convection–diffusion problem. (a) The problem domainABCD and boundary conditions; (b) The solution about a cross-section SS ′ locatedat a distance x from the boundary AD


Consider now the heat equation posed on an infinite domain:

∂φ

∂t− k

∂2φ

∂y2= 0, in Ω := (y, t) | y ∈ (−∞,∞), t ∈ [0,∞) (3.9a)

φ(y, t = 0) = f(y) f(y) := [H(y+ a) − H(y− a)] ,a > 0 (3.9b)

Note that we have initialized the solution with a function f(y) that was used earlierin Eq.(3.6a) to prescribe the Dirichlet boundary condition. The exact solution for theproblem (3.9) can be expressed as follows:

φ(y, t) =1

2

[erf(y+ a√4kt

)− erf

(y− a√4kt

)](3.10)

Clearly, replacing t with (x/u) in Eq.(3.10) we recover the leading term describingthe characteristic layers given by Eq.(3.7). Note that (x/u) is the time required to travela distance x along the characteristic lines. This resemblance is due to the fact that inregions far-away from the domain boundaries the convective and diffusive effects donot interact, i. e.convection just carries the diffusing solution along the characteristiclines [172].

Next, we try to relate the solution of the heat equation with the solution of thediffusion–reaction problem. The statement of the diffusion–reaction problem posedon an infinite domain is:

−kd2φ

dy2+ sφ = sf(y) in Ω := y | y ∈ (−∞,∞) (3.11a)

φ(y) = 0 at y = ±∞ (3.11b)

The exact solution for the above problem can be expressed as follows:

φ(y) =sgn(y+ a)

2

[1− e−ξ|y+a|

]−

sgn(y− a)2

[1− e−ξ|y−a|

](3.12)

where, ξ :=√s/k. Figures 41a and 41b illustrate the solution of the heat equa-

tion given by Eq.(3.10) and the solution of the diffusion–reaction problem given byEq.(3.12) respectively. Clearly these two solutions have distinct profiles. Nevertheless,they share a common trait of possessing parabolic layers, i. e.the first-order deriva-tives in the direction perpendicular to the layers have magnitude O(1/

√k). We refer

to [175] for further details about parabolic and exponential layers.Now we pose the following design problem: Relate s and t such that the parabolic

layers in the solution of the heat equation i. e.Eq.(3.10) and the solution of the diffusion–reaction problem i. e.Eq.(3.12) have the same width.

In the following developments the width of the layer is taken as the distance withinwhich the value of φ varies from 1% to 99% of [max(f(y)) − min(f(y))]. We choosef(y) = H(y) to simplify the algebra. For this choice of f(y) the solution of the heatequation and the diffusion–reaction problem can be expressed as in Eq.(3.13) andEq.(3.14) respectively.

φ(y, t) =1

2

[1+ erf

(y√4kt

)](3.13)

φ(y) =1

2

[1+ sgn(y)

(1+ eξ|y|

)](3.14)


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

← φ(y,t=0) = f(y)

← φ(y,t)

y

(a)

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

← f(y)

← φ(y)

y

(b)

Figure 41: Parabolic layers in the solution of: (a) the heat equation given by Eq.(3.10) usingk = 0.01 and t = 0.1; (b) the diffusion–reaction problem given by Eq.(3.12) usingk = 0.01 and s = 10

√2

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

← 1

2

[1 + erf(

y√4kt

)

]

← 1

2+

sgn(y)

2

[1 − eξ|y|

]

← H(y)

(a)−0.2 −0.15 −0.1 −0.05 0

−0.010

0.01

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1

2

[1 + erf(

y√4kt

)

]→

1

2+

sgn(y)

2

[1 − eξ|y|

]→

H(y) →

(b)

Figure 42: Matching the layers in the solution of the heat equation and the diffusion–reactionproblem. (a) plot domain: [−0.2, 0.2], k = 0.01, t = 0.1 and s := (

√2/t) = 10

√2; (b)

plot domain: [−0.2, 0], the two solutions always meet at a value equal to 0.01

Let y = −y∗ be the distance at which the solutions given by Eq.(3.13) and Eq.(3.14)have a value equal to 1% of [max(H(y)) −min(H(y))], i. e.0.01. Due to the inherent


symmetry of the problem, these solutions at y = y∗ will attain a value equal to 99%of [max(H(y)) −min(H(y))], i. e.0.99. Thus we have,

1

2

[1+ erf

(−y∗√4kt

)]=

1

100=e−ξy

∗

2(3.15)

Solving Eq.(3.15) we get the following equation relating s and t,

st =1

4

[ln(50)

erf−1(49/50)

]2≈√2 ⇒ s ≈

√2

t(3.16)

The above relation between s and t guarantees that the parabolic layers that appearin the solutions of the heat equation and the diffusion–production problem will havethe same width. In Figure 42 using Eq.(3.16) these solutions having the same layerwidth are compared.

We now address the initial objective of quantifying the characteristic layers foundin the singularly perturbed convection–diffusion problem (3.5). Consider a fictitiousreaction coefficient sc and an associated dimensionless element number ωc definedas below.

sc :=

√2u

x, ωc =

sc`2

k(3.17)

where ` is an appropriate element length measure. We have used the substitutiont = (x/u) in Eq.(3.16) to arrive at the expression for sc in Eq.(3.17). Recall that wehave used earlier the same substitution in the solution of the heat equation to recoverthe leading term describing the characteristic layers in the solution of the convection–diffusion problem. We may use this fictitious reaction coefficient sc to relate the char-acteristic layers of the convection–diffusion problem to similar2 parabolic layers of the1d diffusion–production problem. In this sense, the nondimensional element numberωc quantifies the characteristic layers and could be used in the design of stabilizationparameters to control the numerical artifacts about these layers.

Note that the value of sc is a function of x, i. e.sc is inversely proportional to thedistance from the source of the discontinuity along the characteristic lines. In fact thisis how the characteristic layers in the solution of the convection–diffusion problembehave, i. e.their width widens as we move away from the source of the discontinu-ity along the characteristic lines. However from the design point-of-view, a variabledefinition of sc and hence of ωc is inconvenient. This is due to the fact that the charac-teristic lines could be arbitrary curves governed by the velocity field and hence findingthe distance x along these lines need not be straight-forward. Hence we redefine scand ωc using an appropriate element characteristic length `c.

sc :=

√2u

`c, ωc =

sc`2

k(3.18)

3.3 objective characteristic tensors

In this section we present the objective characteristic tensors h and H used in theextension of the HRPG method to higher dimensions. In the developments to follow,

2 in the sense of matched layer widths

3.3 Objective characteristic tensors 75

only the multi-linear block finite elements are considered. Here objectivity is to be un-derstood in the sense popular in tensor calculus, i. e.independence of the method onthe description of the reference frame and admissible node numbering permutationsof the mesh.

Consider the following definition for the element length vectors li :

li := J · li ; Jij :=∂xi∂xj

; l1 := 2(1, 0) ; l2 := 2(0, 1) (3.19)

where J represents the Jacobian matrix of bijective mappings from the local to globalcoordinate systems, xi and xi represent the global and local coordinates respectivelyand li are fixed vectors along the axes of the local frame. Figure 43 illustrates theelement length vectors li obtained at any arbitrary point P(x1, x2) within a 2d bilinearblock finite element. The expression for the vectors li in 2d and at this point P can besimplified to the following:

l1 =1− x22

E12 +1+ x22

E43 ; l2 =1− x12

E14 +1+ x12

E23 (3.20)

where Eab is the edge vector pointing from node a to node b.

P

1 2

3

4

E12

E43

E23

E14l1

l2

(a)

P

1

2

3

4

E12

E43

E23

E14

l1

l2

(b)

Figure 43: Anisotropic element length vectors li obtained at any arbitrary point P(x1, x2)within a 2d bilinear block finite element. The sub-figures (a) and (b) illustrate li

obtained for two admissible global node numbering permutations

Let αi,βi be stabilization parameters calculated along the element length vectorsli and with the following properties: a) (u · li)αi > 0 ∀ i, b) βi > 0 ∀ i and c) onlyscalars and free vectors3 are used in their respective definitions. The definition of theseparameters is delayed until §3.4. The characteristic tensors h and H are calculated as:h := 0.5αili , H := 0.5(βi/|li|)[li ⊗ li]. Thus in 2d the characteristic tensors could beexpressed as follows:

h := α1l1 +α2l2 ; H :=β1

|l1|[l1 ⊗ l1] +

β2

|l2|[l2 ⊗ l2] (3.21)

3 If one is interested only in the magnitude and direction of the vector and does not think of it as situatedat any particular location, then it is called a free vector


Using h, H as defined above we calculate the perturbation ph associated with theHRPG method as described earlier in Eq.(2.6), i. e.

ph := [h + H · ur] ·∇wh (3.22)

ur :=R(φh)

|∇φh|2∇φh; ⇒ ur :=

ur

|ur|=

sgn[R(φh)]|∇φh|

∇φh (3.23)

The definition of h and H given by Eq.(3.21) guarantees the objectivity of the HRPGmethod. Reference frame independence can be verified by the fact that the tensors hand H obey the same tensor transformation rules as any other free tensor associatedwith the problem, e. g.the velocity vector u. Admissible node numbering permutationsonly swap one element length vector with the other (possibly with a change of sign)as shown in Figure 43b. Due to the properties of αi,βi and by their definition, thecharacteristic tensors h and H are invariant with respect to these swaps in li.

Remark 1 : As noted in §2.2 the multidimensional HRPG method could be arrivedat from the FIC equations with an appropriate definition of the characteristic lengthas hfic := h + H · ur.

Remark 2: A figure similar to Figure 43a were presented earlier in [92] (cf. Fig. 3.2,pp. 55), [22] (cf. Fig. 3.4, pp. 215) and [190] (cf. Figure 2, pp. 2205). Therein the elementlength vectors li evaluated at the centroid of the element were used to define a scalarelement size measure. The distinction here is to use these li to arrive at objectivecharacteristic tensors h and H that treat effectively the anisotropy of the finite element.

3.4 stabilization parameters

Except for the modification to include the new dimensionless number introduced in§3.2 that quantifies the characteristic layers, the definition of the stabilization parame-ters αi,βi calculated along the element length vectors li are a direct extension of theircounterparts in 1d summarized in §2.5.6. Following this line, in multi dimensions andalong li we define the following nondimensional element numbers:

γi :=u · li2k

; ωi :=s|li|2

k; σi :=

s|li|2

u · li (3.24)

Following Eq.(3.18), the fictitious reaction coefficient si and the associated dimen-sionless number ωi along li are calculated as follows.

si := maxj6=i

√2|u · lj||lj|2

; ωi :=si|li|2

k(3.25)

Following Eq.(2.78) the stabilization parameters αi along li are calculated as follows.

αi := λi sgn(u · li)max[1−

1

|γi|

], 0

; λi :=1

3(1+√

|σi|)(3.26)

Assuming that the discretization in time is done using the implicit trapezoidal ruleand following Eq.(2.76) we calculate the nonlinear pseudo-reaction coefficient δ asfollows.

δ ≈ 1

θ∆t

‖ φh −φnh ‖∞‖ φh ‖∞ (3.27)

3.5 Examples 77

Following Eq.(2.72) we define the effective convection, diffusion and reaction coef-ficients along li as follows.

ui :=u · li|lj|

−αi|li|s2

−α|li|δ2

; ki := k+αiu · li2

; s := s+ δ (3.28)

Likewise following Eq.(2.73), the effective element dimensionless numbers along li

can be calculated as,

γi :=|ui||li|2ki

; σi :=s|li||ui|

; ωi :=s|li|2

ki(3.29)

Finally, following Eq.(2.74) the stabilization parameters βi along li are calculatedusing the dimensionless numbers γi, σi, ωi and ωi as follows:

βi := max[2

3

(σi + 3

σi + 2

)−

(4

ωi + 4γi

)],[2

3−4

ωi

], 0

(3.30)

The inclusion of the term (2/3) − (4/ωi) in the definition of βi is the only mod-ification from a straight-forward extension to multi dimensions of the definition ofits counterpart in 1d. This expression follows from Eq.(2.55) and the justification isbased on the strategy we employ to treat the numerical artifacts about the character-istic layers— to treat them just like the numerical artifacts about the parabolic layersin the reaction-dominant case.

3.5 examples

In this section we present some examples in 2d for the convection–diffusion–reactionproblem defined by Eq.(2.1). The domain Ω is discretized by considering both struc-tured and unstructured meshes made up of just the bilinear block finite elements.The unstructured meshes are obtained by randomly perturbing the interior nodes ofstructured meshes with coordinates (xi,yi) as follows [60, 128]:

x′i = xi + `1δrand() ; y

′i = yi + `2δrand() (3.31)

where, (x′i,y

′i) represent the corresponding coordinates of the unstructured mesh,

`1, `2 represent the mesh sizes of the structured mesh, δ is a mesh distortion parameterand rand() is a function that returns random numbers uniformly distributed in theinterval [−1, 1]. Figure 44 illustrates two types of unstructured meshes obtained by thisprocedure using a 20× 20 square mesh and the parameter δ = 0.2. In Figure 44a, δ =0.2 was chosen for all the internal nodes of the mesh. Whereas for the nodes adjacentto the boundary in the mesh shown in Figure 44b, the perturbation perpendicular tothe boundary was set to zero. The unstructured meshes obtained using the formerand later techniques are denoted as ‘Type I’ and ‘Type II’ respectively.

3.5.1 Steady-state examples

In this section we illustrate the performance of the HRPG method for the stationaryconvection–diffusion–reaction problem. Unless otherwise specified, in all the exam-ples the following data is considered. The domain Ω is [0, 1]× [0, 1]. Each example is


residual

R(φh) :=∂φh∂t

+ u ·∇(φh) − k∆(φh) + sφh − f(x)

hrpg method

Find φh : [0, T ] 7→ Vh such that ∀wh ∈ Vh0 we have,

a(wh,φh)+∑e

(h ·∇wh,R(φh)

)Ωeh

+( |R(φh)||∇φh|

H ·∇wh,∇φh)Ωeh

= l(wh)

HRPG weight→ wh +

[h +

sgn[R(φh)]|∇φh|

H ·∇φh]·∇wh

definitions


+ u ·∇(φh) − k∆(φh) + sφh − f

φn+1n =(1θ

)φh +

(θ− 1θ

)φnh ; ∆t = tn+1 − tn ; θ ∈ (0, 1)

li := 2J · ei ; Jij :=∂xi∂xj

; si := maxj6=i

√2|u · lj||lj|2

γi :=u · li2k

; σi :=s|li|2

u · li ; ωi :=si|li|2

k

λi :=1

3(1+√|σi|)

; δ ≈ 1

θ∆t

‖ φh −φnh ‖∞‖ φh ‖∞

αi := λi sgn(u · li)max[1−

1

|γi|

], 0

ui :=u · li|li|

−αi|li|s2

−α|li|δ2

; ki := k+αiu · li2

; s := s+ δ

γi :=|ui||li|2ki

; σi :=s|li||ui|

; ωi :=s|li|2

ki

βi := max[2

3

(σi + 3

σi + 2

)−

(4

ωi + 4γi

)],[2

3−4

ωi

], 0

h :=1

2αili ; H :=

1

2

βi

|li|[li ⊗ li]

Box 1: Summary of the HRPG method in multi dimensions and considering the implicit trape-zoidal rule for time integration

3.5 Examples 79

(a) (b)

Figure 44: Unstructured 20× 20 meshes made of bilinear block finite elements. (a) Type I: allinternal nodes of the mesh are perturbed using δ = 0.2. (b) Type II: the perturbationperpendicular to the boundary was set to zero for the boundary-adjacent nodes ofthe mesh. For the rest of the cases δ = 0.2 was chosen.

solved using four meshes, two of which are structured and the remaining two are un-structured. The structured meshes consists of 20× 20 (uniform/square) and 40× 20(rectangular) bilinear elements respectively. The unstructured meshes are obtainedfrom the considered uniform mesh via the two perturbation techniques described ear-lier and illustrated in Figure 44. The obtained solutions are illustrated as surface plotswhose view is described as (θ,ψ), where θ is the azimuthal angle with respect tothe negative y-axis and ψ is the elevation angle from the x–y plane.

Example 1: This is a classical steady-state problem introduced in [22] where theadvection is skew to the mesh with downwind essential boundary conditions. Theproblem data is: u = (5,−9), k = 10−8, s = 0 and f = 0. The boundary conditionsare: φ = 1 on (x = 0,y > 0.7) ∪ (x < 1,y = 1), φ = 0.5 at (x = 0,y = 0.7) andφ = 0 on the rest of the boundary. This problem has exponential boundary layersat the outflow boundary and an internal characteristic layer. Figure 45 illustrates thesolutions obtained by the HRPG method viewed at (20, 20).

Example 2: This problem was studied in [152] wherein a nonuniform rotational veloc-ity field is employed in a rectangular domainΩ := [−1, 1]× [0, 1]. Structured meshes of40× 20 (uniform/square) and 80× 20 (rectangular) bilinear elements are used. The un-structured meshes are obtained from the uniform mesh via the two perturbation tech-niques described earlier. The problem data is: u = 104(y[1− x2],−x[1−y2]), k = 10−4,f = 0, s = 0. The boundary conditions are: φ = 1 on (x < −0.5,y = 0), φ = 0.5 at(x = −0.5,y = 0), φ = 0 on (−0.5 < x 6 0,y = 0) ∪ (x = 1,y) and on the rest of theboundary the Neumann condition n · ∇φ = 0 is imposed. The numerical solution ofthe HRPG method viewed at (20, 20) is shown in Figure 46.

Example 3: This is a plain diffusion–reaction problem. The problem data is: u = (0, 0),k = 10−8, f = 1, s = 1. The homogeneous boundary condition φ = 0 is imposed


(a) (b)

(c) (d)

Figure 45: Example 1, advection skew to the mesh. The solution of the HRPG method viewedat (20, 20) and using (a) a structured 20× 20 mesh, (b) a structured 40× 20 mesh,(c) an unstructured (Type I) 20 × 20 mesh, (d) an unstructured (Type II) 20 × 20mesh.

everywhere. The numerical solution of the HRPG method viewed at (−45, 20) isshown in Figure 47.

Example 4: This is a multidimensional modification of the convection–diffusion–reaction problem studied earlier in §2.5.7.1 and also in [136, 152]. The problem datais: u = (0.01, 0), k = 10−4, s = 4.8 and f = 0. The boundary conditions are: φ = 1.0on (x = 0,y) ∪ (x,y = 0), φ = (3/8) on the rest of the boundary. The value of theelement dimensionless numbers γ1, ω1 are 2.5 and 120 respectively. Recall that forsimilar problem data in 1d (cf. §2.5.7.1 and [136, 152]) the upwind numerical artifactsin the solution of Galerkin method were found to be enhanced in the solution of theSUPG method. The numerical solution of the HRPG method viewed at (120, 20) isshown in Figure 48.

Example 5: This is a uniform advection problem with a constant source term intro-duced in [118]. The problem data is: u = (1, 0), k = 10−8, f = 1, s = 0. The homoge-neous boundary condition φ = 0 is imposed everywhere. The exact solution develops

3.5 Examples 81

(a) (b)

(c) (d)

Figure 46: Example 2, nonuniform rotational advection. The solution of the HRPG methodviewed at (20, 20) and using (a) a structured 40× 20mesh, (b) a structured 80× 20mesh, (c) an unstructured (Type I) 40 × 20 mesh, (d) an unstructured (Type II)40× 20 mesh.

exponential layers at the outlet boundary (x = 1,y) and characteristic boundary layersat (x,y = 0) and (x,y = 1). The numerical solution of the HRPG method viewed at(−45, 20) is shown in Figure 49.

Example 6: This is a non-uniform advection problem with a constant source term in-troduced in [188]. The advection is caused by a unit angular velocity field. Structuredmeshes of 64× 64 (uniform/square) and 128× 64 (rectangular) bilinear elements areused. The unstructured meshes are obtained from the uniform mesh via the two per-turbation techniques described earlier. The problem data is: u = (y,−x), k = 10−6,f = 1, s = 0. The homogeneous boundary condition φ = 0 is imposed everywhere.This problem has a complicated boundary layer. For instance close to the boundary(x,y = 0) and with an increase in x, these layers gradually vary from being parabolicto exponential while maintaining a constant profile height φ(x,y = 0) ≈ (π/2) (awayfrom the corners). Close to the boundary (x = 1,y) the solution profile is approxi-mately φ(x = 1,y) ≈ (π/2)− 2 tan−1(y). The numerical solution of the HRPG methodviewed at (−200, 20) is shown in Figure 50.

Example 7: This is a uniform advection problem with a discontinuous source termintroduced in [132]. The problem data is: u = (1, 0), k = 10−8, f(x 6 0.5,y) = 1,f(x > 0.5,y) = −1, s = 0. The homogeneous boundary condition φ = 0 is imposedeverywhere. Structured meshes of 30× 30 (uniform/square) and 60× 30 (rectangular)


(a) (b)

(c) (d)

Figure 47: Example 3, a reaction–diffusion problem The solution of the HRPG method viewedat (−45, 20) and using (a) a structured 20×20mesh, (b) a structured 40×20mesh,(c) an unstructured (Type I) 20 × 20 mesh, (d) an unstructured (Type II) 20 × 20mesh.

bilinear elements are used. The unstructured meshes are obtained from the uniformmesh via the two perturbation techniques described earlier. The numerical solutionof the HRPG method viewed at (−10, 20) is shown in Figure 51.

3.5.2 Transient examples

Here we illustrate the performance of the HRPG method for the transient 2d pureconvection problem. Only uniform bilinear finite elements are used here. Both ofthe examples presented here deal with the advection of solid bodies modeled withappropriate density functions. These problems are frequently used as test cases foradvection algorithms demonstrating their treatment of dispersive oscillations and theoverall solution accuracy.

Example 8: This is a test case introduced in the ERCOFTAC document [24]. A circu-lar scalar bubble is initially positioned at the bottom of a square domain in a fixedconstant velocity field directed at 45 toward the top right of the domain. The prob-lem data is: u = (0.5, 0.5), k = 10−30, s = 0 and f = 0. The domain Ω := [0, 3]× [0, 3] is

3.5 Examples 83

(a) (b)

(c) (d)

Figure 48: Example 4, a convection–diffusion–reaction problem with a dominant reaction term.The solution of the HRPG method viewed at (120, 20) and using (a) a structured20× 20 mesh, (b) a structured 40× 20 mesh, (c) an unstructured (Type I) 20× 20mesh, (d) an unstructured (Type II) 20× 20 mesh.

discretized by a uniform mesh of 300× 300 bilinear elements. The time integration isdone using the implicit midpoint rule and is advanced at a time step of 0.005 seconds.This corresponds to an element CFL number of 0.25. Define a radius R = 0.25, an arbi-trary position vector r := (x,y) ∈ Ω and a specific position vector rc := (0.5, 0.5) ∈ Ω.The initial solution can then be expressed as follows:

φ(r, t = 0) = H(R− |r − rc|) (3.32)

where H() is the Heaviside function defined earlier in Eq.(3.6b) and rc is the centerof the circular scalar bubble. The initial solution viewed at (40, 20) is shown inFigure 52a (elevation plot) and Figure 52c (contour plot). The Dirichlet boundarycondition φ = 0 is imposed at the inlet boundaries. The numerical solution of theHRPG method at time t ∈ 1, 2, 3, 4 seconds and viewed at (40, 20) is shown inFigure 53 (elevation plots) and Figure 54 (contour plots).

Example 9: This is a standard benchmark problem introduced in [125] that simulatesthe advection of a solid body subjected to a constant angular velocity field. The solidbody is modeled with a scalar density function that has three shapes, viz. a slotted


(a) (b)

(c) (d)

Figure 49: Example 5, uniform advection with a constant source term. The solution of theHRPG method viewed at (−45, 20) and using (a) a structured 20× 20 mesh, (b) astructured 40× 20 mesh, (c) an unstructured (Type I) 20× 20 mesh, (d) an unstruc-tured (Type II) 20× 20 mesh.

cylinder, a cone and a sinusoidal hump. The classical problem with just the slottedcylinder revolving about the center of the domain was proposed by Zalesak in theseminal paper [191] that extended the FCT method to multi dimensions. The problemdata is: u = (0.5−y, x− 0.5), k = 10−30, s = 0 and f = 0. The domainΩ := [0, 1]× [0, 1]is discretized using 200× 200 uniform bilinear elements. The time integration is doneusing the implicit midpoint rule and is advanced at a time step of 0.001 seconds. Thiscorresponds to a maximum element CFL number of 0.1. Define a radius R = 0.15, anarbitrary position vector r := (x,y) ∈ Ω and a specific position vector ra := (xa,ya) ∈Ω for some chosen point a. The initial solution can then be expressed as follows:

φ(r, t = 0) = H(R− |r − r1|)[1− H(0.025− |x− x1|)H(0.85− y)

]+

1− min|r − r2|R

, 1+1

4

[1+ cos

(πmin

|r − r2|R

, 1)] (3.33)

where H() is the Heaviside function defined earlier in Eq.(3.6b), r1 = (0.5, 0.75),r2 = (0.5, 0.25) and r3 = (0.25, 0.5) are the position vectors corresponding to the cen-ter of the slotted cylinder, the cone and the sinusoidal hump respectively. The initial

3.5 Examples 85

(a) (b)

(c) (d)

Figure 50: Example 6, non-uniform advection with a constant source term. The solution ofthe HRPG method viewed at (−200, 20) and using (a) a structured 64× 64 mesh,(b) a structured 128× 64 mesh, (c) an unstructured (Type I) 64× 64 mesh, (d) anunstructured (Type II) 64× 64 mesh.

solution viewed at (−20, 20) is shown in Figure 52b (elevation plot) and Figure52d (contour plot). The Dirichlet boundary condition φ = 0 is imposed at the in-let boundaries. Under the considered velocity field the initial solution completes afull revolution in 2π seconds. The numerical solution of the HRPG method at timet = (π/2),π, (3π/2), 2π seconds and viewed at (−20, 20) is shown in Figure 55

(elevation plots) and Figure 56 (contour plots).

3.5.3 Discussion

The HRPG method proposed here can be understood as the combination of up-winding plus a discontinuity-capturing operator. Also the discontinuity-capturing


(a) (b)

(c) (d)

Figure 51: Example 7, uniform advection with a discontinuous source term. The solution ofthe HRPG method viewed at (−10, 20) and using (a) a structured 30× 30 mesh,(b) a structured 60× 30 mesh, (c) an unstructured (Type I) 30× 30 mesh, (d) anunstructured (Type II) 30× 30 mesh.

term has the canonical form of the shock-capturing diffusion, i. e.it is proportionalto (|R(φh)|/|∇φh|). Nevertheless the finer structure of the HRPG method is distinctfrom the existing shock-capturing Petrov–Galerkin methods in the literature (cf. Table2 in chapter 2). The distinction is that the upwinding provided by the characteristictensor h is not streamline and the discontinuity capturing provided by the character-istic tensor H is neither isotropic nor purely crosswind.

It is clearly seen from the steady-state examples presented in the previous sectionthat for structured meshes (both square and rectangular bilinear elements) the HRPGmethod reproduces a crisp resolution of the layers in the numerical solution. The goodperformance on rectangular elements (here considered with an aspect ratio of 2:1) isdue to the anisotropic treatment of the stabilization terms involving the characteristictensors h and H. The solutions obtained by the HRPG method for the transient 2d

advection examples advocate its good treatment of dispersive oscillations withoutcompromising the solution-accuracy (cf. Figures 53 and 55). Also the symmetry of theinitial data is well maintained (cf. Figures 54 and 56). Recall that the time integrationwas performed by the implicit midpoint rule which is a symplectic time integrator[76]. This choice was made to single-out the treatment of the geometrical symmetryin the initial data by the HRPG method.

Clearly on unstructured meshes we do not attain the same layer resolution qualityas is obtained on the corresponding structured meshes. However the parabolic lay-ers (characteristic and reactive layers) are captured satisfactorily. About the exponen-

3.5 Examples 87

(a) (b)

(c) (d)

Figure 52: Initial data for the transient 2d advection examples. (a) Example 8, elevation plotviewed at (40, 20), (b) Example 9, elevation plot viewed at (−20, 20), (c) Exam-ple 8, contour plot, (d) Example 9, contour plot.

tial layers some overshoots and undershoots are observed using Type I unstructuredmeshes. These unwanted localized artifacts are conspicuous in the solutions of exam-ple 5 (Figure 49c) and example 6 (Figure 50c) suggesting that there is room for furtherimprovement of the method. Nevertheless using Type II unstructured meshes wherein the random perturbation of the mesh nodes perpendicular to the domain bound-ary is set to zero, these unwanted artifacts about the exponential layers are greatlyreduced.

Figure 51 illustrates another shortcoming of the HRPG method that is conspicuouseven when structured meshes are used. On one half of the domain (here the sourceterm is positive) the obtained solutions have crisp layer resolutions, whereas in theremaining half (here the source term is negative) the numerical solution appears tobe over-damped and even negative near the corners of the outlet boundary. This isa shortcoming suffered by all the shock-capturing techniques designed within thePetrov–Galekin framework (see Codina’s monograph [31]) that rely on the canoni-


(a) (b)

(c) (d)

Figure 53: Example 8, transient pure convection skew to the mesh. The solution of the HRPGmethod viewed at (40, 20) and at time (a) t = 1 s, (b) t = 2 s, (c) t = 3 s, (d) t = 4s.

cal strategy of adding a positive shock-capturing diffusion. The following exampleillustrates why the aforesaid strategy fails to address this shortcoming.

Example 10: Consider a unit domain Ω := [0, 1]× [0, 1] and the following problemdata: u = (1, 0), k = 10−8, s = 0 and f = −1. The Dirichlet boundary conditionsare: φ = 1 on (x = 0,y > 0) ∪ (x,y = 1) and φ = 0 on the rest of the boundary.The domain Ω is discretized using a structured mesh of 20 × 20 (uniform/square)bilinear elements. In the interior of the domain the exact solution has the profile ofa flat surface with a slope of −1. Along the boundaries (x,y = 0) and (x,y = 1) theexact solution develops characteristic boundary layers and as a consequence withinthe width of these characteristic layers and near the corners of the outlet boundary(x = 1,y), exponential layers are formed. Hence the solution of the plain Galekin FEMwill be corrupted with global oscillations. The solutions obtained by the SUPG andthe HRPG method are shown in Figure 57.

Note that the undershoots and overshoots in the solution of the SUPG method isidentical across both characteristic layers (cf. Figures 57a and 57c). This is in agreementwith the reasoning made in §3.1 related to the numerical artifacts across characteristiclayers, i. e.unlike in the reaction-dominant case where it is the numerical solution

3.5 Examples 89

(a) (b)

(c) (d)

Figure 54: Example 8, transient pure convection skew to the mesh. The contour plots of thesolution of the HRPG method at time (a) t = 1 s, (b) t = 2 s, (c) t = 3 s, (d) t = 4 s.

that undergoes the 1d mass type averaging, in the convection-dominant case it isthe derivatives of the numerical solution that undergoes the same. Thus, the Gibbsphenomenon across the characteristic layers in the later case is proportional to thevariation in the derivatives of the solution across these layers. In other words for thecurrent problem it is the slope of φh and not the actual value of φh on the boundarythat determines the observed artifacts. It can be clearly seen in Figure 57c that anymethod, that relies on the canonical strategy of adding a positive shock-capturingdiffusion, will not be able to recover (near the boundary (x,y = 0)) the nodally exactinterpolant from the initial SUPG solution. On the other hand, note that the artifactsnear the boundary (x,y = 1) has a profile similar to the one that would have been


(a) (b)

(c) (d)

Figure 55: Example 9, rotation of solid bodies. The solution of the HRPG method viewed at(−20, 20) and at time (a) t = (π/2) s i. e.after a quarter-revolution, (b) t = π si. e.after a half-revolution, (c) t = (3π/2) s, i. e.after three quarters of a revolution(d) t = 2π s i. e.after a full-revolution.

observed for the L2 projection of the exact solution onto the finite element space. It isfor this reason that the aforesaid strategy succeeds in capturing these layers.

Obviously tailor-made solutions exist to treat this shortcoming. For instance, onesuch trick that recovers crisp resolution of these layers for the HRPG method and forthe current problem (example 10) is to reverse the sign of the stabilization parameterβ (along the y-axis) for all elements containing the boundary section (x > 0.5,y = 0),thus enforcing a negative shock-capturing diffusion for these elements. Unfortunatelyit is difficult to generalize these tailor-made tricks to an arbitrary situation. An alter-native would be to change the strategy to the one which directly treats the cause ofthe Gibbs phenomenon for both the reactive and characteristic layers4—Design theweights of a Petrov–Galerkin FEM such that the typical 1d mass type averaging inthe Galerkin FEM (cf. Eq.(3.4)) be lumped in the regions across the layers. Researchin this line is still under development and we delay its introduction to future works.

Remark: Fortunately, this idea which was born to treat this shortcoming in theconvection–diffusion–reaction problem, has opened door to a class of higher-order

4 this idea is a fruit of the discussions with Prof. Ramon Codina

3.6 Conclusions 91

(a) (b)

(c) (d)

Figure 56: Example 9, rotation of solid bodies. The contour plots of the solution of the HRPGmethod at time (a) t = (π/2) s i. e.after a quarter-revolution, (b) t = π s i. e.after ahalf-revolution, (c) t = (3π/2) s, i. e.after three quarters of a revolution, (d) t = 2π si. e.after a full-revolution.

compact Petrov–Galerkin FEM effective for the Helmholtz problem. The design ofsuch a Petrov–Galerkin FEM and its applications to the Helmholtz equation is thesubject matter of chapter 5.

3.6 conclusions

We have developed a multi dimensional extension of the HRPG method presentedearlier in chapter 2 for the 1d convection–diffusion–reaction problem. As the charac-teristic internal/boundary layers found in the convection-dominant case are a unique


−0.25

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a)−0.25

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.25

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

φ h

SUPGHRPGInterpolant

cross section at x = 0.05



(c)

Figure 57: Example 10, uniform advection with a negative source term. The solution obtainedon a uniform 20× 20 mesh viewed at (45, 0) and using (a) the SUPG method,(b) the HRPG method. (c) Comparison of the nodally exact interpolant at threedifferent cross sections with the numerical solution obtained by the SUPG and theHRPG method.

feature of the solution in higher dimensions, they do not have any counterparts in1d. Hence, a straight-forward extension of the stabilization parameters of the HRPGmethod derived for the 1d case will not be efficient to resolve these parabolic layers.

The numerical artifacts that are formed across the parabolic layers are usually man-ifested as the Gibbs phenomenon. The strategy we employ to treat the artifacts aboutthe characteristic layers is to treat them just like the artifacts found across the paraboliclayers in the reaction-dominant case. This is done by relating the characteristic lay-ers in the convection–diffusion problem to the parabolic layers formed in a fictitiousdiffusion–reaction problem. The fictitious reaction coefficient in the later problem isdesigned such that the parabolic layers in both the problems have the same width.Using this fictitious reaction coefficient, we present a nondimensional element num-

3.6 Conclusions 93

ber that quantifies these characteristic layers. By quantification we mean that it shouldserve a similar purpose in the definition of the stabilization parameters as the elementPeclet number does for the exponential layers.

Although the structure of HRPG method in 1d is identical to the consistent approx-imate upwind Petrov–Galerkin method [68], in multi dimensions the former methodhas a unique structure. The distinction is that in general the upwinding is not stream-line and the discontinuity-capturing is neither isotropic nor purely crosswind. In thisline, we present anisotropic element length vectors li and using them objective char-acteristic tensors associated with the HRPG method are defined. Only the multilin-ear block finite elements are considered in this study. Except for the modification toinclude the new dimensionless number that quantifies the characteristic layers, thedefinition of the stabilization parameters αi,βi calculated along the element lengthvectors li are a direct extension of their counterparts in 1d summarized earlier in§2.5.6.

Finally, several steady-state and transient examples are presented that throws lighton the good performance of the proposed method.

Part II

H E L M H O LT Z P R O B L E M

Most of what we learn, we learn indirectly or by ’head-fake’.

— Randy Pausch.

4A L P H A - I N T E R P O L AT I O N O F F E M A N D F D M

4.1 introduction

In this chapter we study the Helmholtz equation given by R(φ) := f(x)+∆φ+ξ2φ = 0

and subjected to Dirichlet boundary conditions. The solution φ to this equation isoscillatory and ξ is the wave number (spatial frequency) of φ. If λm is an eigenvalueof the operator −∆, then for ξ 6=

√λm the problem has a unique solution. On the

contrary, i. e.for ξ =√λm the problem is indefinite. In this case if the equation and

the Dirichlet boundary conditions are homogeneous then we end up in a differentialeigenvalue problem. It follows that the solution is not unique and can be representedas a scalar multiple of the eigenfunction corresponding to each eigenvalue. Let λhmrepresent an eigenvalue of the problem after any appropriate discretization. Unlikethe set of eigenvalues λm which is infinite, the set λhm is finite and its dimensionis equal to that of the discrete space. Thus when the wave number ξ →

√λhm the

discrete problem tends to be indefinite. This case is usually referred to as the case ofdegeneracy and here the discrete problem is ill-conditioned.

As the current problem admits a variational principle, naturally, discretization meth-ods based on variational formulations viz. the Galerkin and the Trefftz–Galerkintype methods have been preferred to other methods. The Galerkin type methods aredomain-based wherein the integral statement involves only the weak form of the gov-erning differential equation and the sub-space of test-functions are assumed to satisfya priori the kinematic compatibility and essential boundary conditions. The Trefftz–Galerkin type methods are boundary-based and are formulated using the reciprocalprinciple wherein the integral statement involves only the kinematic compatibilityand essential boundary conditions of the problem and the sub-space of test-functionsare assumed to satisfy a priori the governing differential equation [27, 41].

In the context of the Galerkin type methods, the finite element method (FEM) is apowerful technique to systematically generate subspaces of test-functions (classicallypiecewise polynomial spaces). Some of the earlier works on the use of FEM for the nu-merical solution of the Helmholtz equation can be found in [5, 6, 9, 47, 107, 108, 130]and the references cited therein. In [5, 107] error estimates were given for the asymp-totic (ξ2` assumed sufficiently small) and pre-asymptotic (ξ` assumed sufficientlysmall) cases respectively. It was shown that for the discrete problem the LBB1 constantcan be expressed as γh = min|λhm − ξ2|/λhm [47]. Thus for the continuous problem(visualized as ` → 0) the LBB constant can be expressed as γ = min|λm − ξ2|/λm

which in an average sense implies that γ is inversely proportional to the wavenumberξ, i. e.γ ∝ ξ−1 [47, 107]. Thus for high wavenumbers and for the case of degeneracy(ξ →

√λhm) the LBB constant for the discrete problem tends to be small which in

1 Ladyzhenskaya-Babuska-Brezzi constant

97

98 4 alpha-interpolation of fem and fdm

turn leads to a loss of stability. The loss of stability with respect to an increase in thewavenumber ξ is called the pollution effect which is impossible to avoid completely[9]. Nevertheless the pollution effect can be controlled unlike the loss of stability forthe case of degeneracy where it is out of control.

Several stabilization methods were developed to control the pollution effect ofthe Galerkin FEM. The Galerkin least squares (GLS) method was extended to theHelmholtz equation in [79, 181]. In [79] the extension of the Galerkin gradient leastsquares (GGLS) method for the current problem was also studied. In order to re-tain stability for problems that involve the physics of both the convection–diffusion–reaction and Helmholtz equations the GLSGLS method was proposed [80]. Followingthe framework of the Generalized Finite Element Methods (GFEM) which were firstintroduced in [8] in a variational setting, the Quasi-Stabilized FEM (QSFEM) wasproposed in [10]. To be precise, within an algebraic setting a 9-node interior stencilwas designed such that the pollution effect is asymptotically minimal, thus leading tominimal phase error for arbitrary wave direction in 2d. The partition of unity method(PUM) was proposed in [7, 131] by which conforming subspaces of higher regularitycan be generated out of a set of local approximation spaces. These local approxima-tion spaces could be designed to include a priori knowledge about the local behaviorof the solution. Recently, following the framework of PUM, a locally enriched FEMwas proposed in [121] wherein it was shown that the Bessel functions of the first kindcould be used to enrich the finite element space instead of the plane waves (as isdone in PUM). Another stabilization approach consists of enriching the classical fi-nite element spaces by bubble functions. Following this line the residual-free bubbles(RFB) method was extended to the Helmholtz equation in [67]. Another bubble-basedmethod is the nearly optimal Petrov–Galerkin method (NOPG) presented in [13]. Acomparison of the RFB and NOPG methods for the Helmholtz equation was donein [78]. Recently another GFEM was proposed in [173] in which the classical FEMis enriched by plane waves pasted into the finite element basis at each mesh ver-tex by the PUM. Also, this method allows the use of Cartesian meshes which mayoverlap the boundaries of the problem domain. This GFEM was further developedin [174] wherein the effects of using alternative handbook functions and mesh typesis addressed. Based on the variational multiscale (VMS) method several stabilizationmethods were proposed, viz. the sub-grid FEM [159], the two sub-grid scale (SGS)models presented in [28], the residual-based FEM (RBFEM) [160] and more recently,the algebraic subgrid FEM (ASGS) [74] and the SGS-GSGS method [86]. Followingthe more general VMS method wherein the subscales are not modeled as bubbles, theRBFEM method also includes the residuals on the inter-element boundaries while re-taining the sparsity of the Galerkin method. As in the GLSGLS method, the SGS-GSGSmethod attempts to stabilize the advection–diffusion–reaction/production problemand is designed to be nodally exact in 1d. Within the framework of the discontinuousGalerkin (DG) method, the discontinuous enrichment method (DEM) was proposed[57, 58] wherein the classical finite element spaces are enriched (as in bubble-basedmethods) via a set of local approximation spaces (as in the PUM-based methods).In the DEM, the continuity of the enrichment across element boundaries is enforcedweakly by Lagrange multipliers (unlike the PUM-based methods) and it need notvanish at the element boundaries (unlike the bubble-based methods). Another DGmethod is presented in [3] wherein the continuity of the finite element spaces across

4.1 Introduction 99

the element edges is relaxed and weakly enforced via two penalty parameters cor-responding to possible jumps of the solution field and its gradient. These penaltyparameters are designed to minimize the pollution error. Following the ideas of theformer DG method [3], another discontinuous FEM was proposed in [129] (thereincalled as the DGB method). In the DGB method the classical finite element spacesare enriched via bubbles that are allowed to be discontinuous across subgrid patches.Following the DG method in [3] the continuity of the bubble spaces across interiorpatch boundaries is enforced weakly via two penalty parameters corresponding topossible jumps of the solution field and its gradient. Again, these penalty parametersare designed to minimize the pollution error. Nodally exact Ritz discretization of the1d diffusion-absorption/production equations via variational finite calculus (FIC) andmodified equation methods using a single stabilization parameter were presented in[59]. The Galerkin projected residual (GPR) method for the Helmholtz equation waspresented in [51]. A survey of finite element methods for time-harmonic acoustics isdone in [77].

Due to the abstractness in the definition of the QSFEM, it is often labeled as a finitedifference method. Nevertheless, it provides solutions that are sixth-order accurate,i. e.O

((ξ`)6

)which is the best one can get on any compact stencil. Recently, a quasi-

optimal Petrov–Galerkin (QOPG) method using bilinear finite elements was proposedin [128] that recovers the QSFEM stencil on square meshes. In the QOPG method theGalerkin FEM weights are perturbed by a quadratic bubble function defined overthe macro-element. The parameters multiplying the bubble perturbations are foundby solving local optimization problems involving a functional of the local truncationerror. Later, following this line, a quasi-optimal finite difference method on genericunstructured meshes was proposed in [60].

Within the framework of the finite difference methods, several fourth-order com-pact schemes obtained through a generalization of the fourth-order Padé approxima-tion were studied in [81, 168]. Following this line, two new FDMs were proposed in[165] that achieves sixth and eight-order accuracy respectively using a five-point (andhence non-compact) stencil in 1d. In [122] a new FDM with improved accuracy wasproposed by modifying the central difference scheme (i. e.the classical FDM) by replac-ing the weight multiplying the central node with an optimal expression that used theBessel’s function of the first kind. The FLAME method was proposed in [184] that ex-ploits the use of local approximating functions to define higher-order finite differenceschemes on a chosen stencil. In particular on a compact stencil a sixth-order accuratescheme for the Helmholtz equation can be derived using the FLAME method. Sixth-order accurate FD schemes on a compact stencil for the Helmholtz equation were pro-posed in [135, 169, 176]. An alternate approach to derive FDMs is the global methodof differential quadrature (DQ) [15]. Following this line, a polynomial-based DQ anda Fourier expansion-based DQ were derived for the Helmholtz equation in [166]. Ashigher-order polynomial or sinusoidal interpolation functions are employed, thesemethods reduce the restriction on the mesh resolution to the Nyquist limits, i. e.therule of thumb for these methods is to provide at least two elements per wavelength.

Some of the earlier works in the context of the Trefftz–Galerkin type methods couldbe found in the seminal papers [87, 112, 162, 192]. A treatise on Trefftz type methodscan be found in [88, 113]. Specifically, the Trefftz type methods were used for theHelmholtz equation in [25, 69, 126, 133, 170]. The case of degeneracy, i. e.ξ → λhm, is


considered for the first time in [126] and the error asymptote of the solution by theTrefftz method is given.

In this chapter we present some observations and related dispersion analysis ofa domain-based fourth-order compact scheme for the Helmholtz equation. In otherwords, the phase error of the numerical solution and the local truncation error ofthis scheme for plane wave solutions diminish at the rate O

((ξ`)4

). The focus is

on the approximation of the Helmholtz equation in the interior of the domain usingcompact stencils. The scheme consists in taking the alpha-interpolation of the Galerkinfinite element method (FEM) and the classical finite difference method (FDM). Thisscheme has its origins in an old idea which marks the point of departure: to replacethe consistent mass matrix M in the Galerkin FEM by a higher-order mass matrixM0.5 := (M + ML)/2, where ML is the lumped mass matrix. This idea was proposedindependently for eigenvalue problems by Goudreau [70, 71] and Ishihara [110]. Inthe later work the matrix M0.5 was denominated as the mixed-mass matrix and asa concluding remark the generalized mixed mass (GMM) scheme was proposed asan extension to the MM scheme where an α-interpolation of the mass matrices isdone, i. e.Mα := αM+ (1−α)ML. This GMM scheme was later baptized as the alpha-interpolation method (AIM) [140] and was extended to the hollow waveguide analysisin [109] and the Schrodinger equation in [139]. For the simple 1d case our schememimics the AIM and in 2d making the choice α = 0.5 we recover the generalizedfourth-order compact Padé approximation [81, 168] (therein using the parameter γ =

2).The chapter is organized as follows. In Section 4.2 we present the statement of the

Helmholtz equation viewed as a diffusion–production problem. This is done only tofacilitate future assimilation of ideas towards a generic method that would aim at sta-bilizing problems that involve the physics of both the convection–diffusion–reactionand Helmholtz equations. In Section 4.3 we present the analysis of the problem in1d. The expressions for the numerical solution of our scheme and its relative phaseerror are given considering a generic definition of the parameter α given as a seriesexpansion in terms of (ξ`). A numerical example is given that illustrates not onlythe approximation properties of our scheme but also throws light on possible en-counters with the zones of degeneracy. In Section 4.4 we present the 2d analysis ofa nonstandard compact stencil which results from a two-parameter scheme whereinα-interpolations of the diffusion and production terms are done independently andit can model several methods (including QSFEM). This nonstandard compact stencilhas an additional structure that reduces its abstractness and hence could be exploitedfor the extension of this stencil to unstructured meshes (cf. Section 4.4.5). We follow[10] for the analysis of this stencil and its performance on square meshes is comparedwith that of the quasi-stabilized FEM (QSFEM) [10]. Just like in 1d, we try to expressthe numerical solution of this stencil in 2d considering generic definitions of the pa-rameters given as a series expansion in terms of (ξ`). Using this expression for thenumerical solution, the expressions for the relative phase and local truncation errorsare given. In particular for our scheme, i. e.the α-interpolation of the FEM and FDMstencils an optimal expression for the parameter α is given. The dispersion plots in2d and related discussion are done in Section 4.4.6. Some examples are presentedin Section 4.4.7 which illustrate the pollution effect through convergence studies inthe L2 norm, H1 semi-norm and the l∞ Euclidean norms. Finally in Section 4.5 we

4.2 Problem statement 101

remark on the extension of our scheme to unstructured meshes and arrive at someconclusions.

4.2 problem statement

The statement of the multidimensional Helmholtz equation subjected to Dirichletboundary conditions is as follows:

R(φ) := k∆φ+ sφ+ f(x) = 0 in Ω (4.1a)

φ = φp on ΓD (4.1b)

where k > 0, s > 0 are the diffusion and production coefficients respectively, f(x) isthe source and φp is the prescribed value of φ at the Dirichlet boundary. When s < 0the Eq.(4.1) represents the diffusion-reaction problem that models the mass transferprocesses with first-order chemical reactions and wherein s represents the reactioncoefficient.

The variational statement of the problem (4.1) can be expressed as follows: Findφ ∈ V such that ∀w ∈ V0 we have,

a(w,φ) = l(w) (4.2a)

a(w,φ) :=∫Ω

(k∇w ·∇φ− swφ) dΩ (4.2b)

l(w) :=

∫Ω

wf(x) dΩ (4.2c)

where, V := w : w ∈ H1(Ω) and w = φp on ΓD and V0 := w : w ∈ H1(Ω) and w =

0 on ΓD. The statement of the Galerkin method applied to the weak form (4.2) of theproblem is: Find φh ∈ Vh such that ∀wh ∈ Vh0 we have,

a(wh,φh) = l(wh) (4.3)

where Vh ⊂ V is a subspace obtained via any appropriate discretization. Discretiza-tion of the space by finite elements will lead to the approximation φh = NaΦa andEq.(4.3) reduces into the following system of equations.

[kD − sM

]Φ = f (4.4a)

Dab =

∫Ω

∇Na ·∇Nb dΩ, Mab =

∫Ω

NaNb dΩ, fa =

∫Ω

Naf(x) dΩ

(4.4b)

4.3 analysis in 1d

4.3.1 Introduction

In this section we study the homogeneous Helmholtz equation in 1d subjected toDirichlet boundary conditions. The problem (4.1) in 1d can be written as:

kd2φdx2

+ sφ = 0 in Ω (4.5a)

φ(x = 0) = Φl ; φ(x = L) = Φr on ΓD (4.5b)


where L is the length of the 1d domain and Φl,Φr are the Dirichlet boundary data atthe left and right domain boundaries respectively. The solution to Eq.(4.5) when s > 0is harmonic and is expressed as:

φ(x) =Φl sin(ξoL− ξox) +Φr sin(ξox)

sin(ξoL)(4.6)

where ξo :=√s/k is the angular wave number. We also list the eigenvalues of this

problem which can be expressed as λm := (mπ/L)2 ∀ m ∈ 1, 2, 3, . . .. The elementcontributions to the matrices given in Eq.(4.4) using 2-node linear finite elements are,

De =1

`

[1 −1

−1 1

]; Me =

`

6

[2 1

1 2

](4.7)

where ` is the corresponding element length. If the discretization is uniform the equa-tion stencil for the problem (4.5a) corresponding to each interior node can be ex-pressed as follows,

(k

`

)(−Φi−1 + 2Φi −Φi+1) −

(s`

6

)(Φi−1 + 4Φi +Φi+1) = 0 (4.8)

If the mass matrix M is lumped then the equation stencil corresponding to any interiornode can be written as follows.

(k

`

)(−Φi−1 + 2Φi −Φi+1) − s`Φi = 0 (4.9)

This is also the stencil we get using the classical finite difference method2 (FDM).

4.3.2 α-Interpolation of the Galerkin-FEM and the classical FDM

Define a free parameter α and consider the α-interpolation of the stencils obtained bythe Galerkin FEM and the classical FDM methods for the problem (4.5):

(1−α)

[(k

`

)(−Φi−1 + 2Φi −Φi+1) −

(s`

6

)(Φi−1 + 4Φi +Φi+1)

]

+α

[(k

`

)(−Φi−1 + 2Φi −Φi+1) − s`Φi

]= 0

(4.10a)

⇒(k

`

)(−Φi−1 + 2Φi −Φi+1) − (1−α)

(s`

6

)(Φi−1 + 4Φi +Φi+1)

−αs`Φi = 0

(4.10b)

⇒(k

`−α

s`

6

)(−Φi−1 + 2Φi −Φi+1) −

(s`

6

)(Φi−1 + 4Φi +Φi+1) = 0 (4.10c)

Remark: In 1d we can arrive at the above equations through an alternative argument:Consider the Galerkin FEM method using the α-interpolated mass matrix Mα. Thelater argument leads to the AIM. A particular case (taking α = 0.5) is the mixed-mass(MM) scheme proposed by Ishihara applied to the Helmholtz equation [110]. The

2 by classical FDM we refer to the central difference scheme


mixed-mass matrix (M0.5) was earlier referred to as the higher-order-mass matrix byGoudreau [70]. In 1d a stencil equivalent to the MM scheme will be obtained usingthe compact fourth-order Padé approximation to problem (4.5) [81, 168].

We can guess that a solution to Eq.(4.10) takes the form Φi := φ(xi) = exp(iξhxi).Substituting this solution into Eq.(4.10) and defining λ := exp(iξh`) we get the char-acteristic equation of the stencil:

λ2 − 2

(6− (2+α)ω

6+ (1−α)ω

)λ+ 1 = 0 (4.11)

where ω := (s`2/k) = (ξo`)2 is a dimensionless element number. The solution to

Eq.(4.11) can be expressed as follows.

λ := eiξh` = fα±

√(fα)2 − 1 = fα± i

√1− (fα)2 ; fα :=

(6− (2+α)ω

6+ (1−α)ω

)(4.12)

Note that if |fα| 6 1 then the solution given by Eq.(4.12) is real (i. e.ξh ∈ R). Thissolution can be expressed as a series expansion in terms of ω as follows:

ξh` = cos−1(fα) = cos−1(6− (2+α)ω

6+ (1−α)ω

)=√ω

[1−

(2α− 1

24

)ω

+(20α2 − 20α+ 9

1920

)ω2 +

(280α3 − 420α2 + 378α− 103

193536

)ω3 +O

(ω4)]

(4.13)

Should the expression for α be written as a generic series expansion in terms of ωgiven by α =

∑∞m=0 amω

m, then the solution ξh can be written as shown below.

ξh` =√ω

[1+

(1− 2a024

)ω+

(20a20 − 20a0 + 91920

+a112

)ω2

+(280a30 − 420a20 + 378a0 − 103

193536+

(2a0 − 1)a148

+a112

)ω3 +O

(ω4)]

(4.14)

where am are coefficients independent of ω. The relative phase error of the abovesolution can be expressed as shown below.

ξh − ξoξo

=ξh`−

√ω√

ω=

[(1− 2a024

)ω+

(20a20 − 20a0 + 91920

+a112

)ω2 +O

(ω3)]

(4.15)

Note that for the choice a0 = 1/2, the relative phase error diminishes at the rateof O

(ω2)

or equivalently O((ξo`)

4). Further, making the choice a1 = −1/40, the

relative phase error now diminishes at the rate of O(ω3)

or equivalently O((ξo`)

6).

Fortunately in 1d it is possible to choose α such that the solution given by Eq.(4.12) benodally exact (i. e.ξh` = ξo` =

√ω). The expression for α that reproduces this effect,

say αe, can be written as follows:

fαe = cos(ξo`) = cos(√ω) ⇒ αe =

6

ω−

(2+ cos(

√ω)

1− cos(√ω)

)(4.16)

The optimal parameter αe can be expressed as a series expansion in terms of ωas shown in Eq.(4.17). Truncating the series up to the first n terms would yield ascheme whose relative phase error diminishes at the rate of O

(ωn+1

)or equivalently

O((ξo`)

2n+2).

αe ≈1

2−ω

40−ω2

1008−

ω3

28800−

ω4

887040−

691ω5

19813248000+O

(ω6)

(4.17)


4.3.3 Dispersion plots in 1d

In this section we consider α ∈ 0, 1, 0.5,αe and study their dispersion plots. Thesubscripts c, l,m are flags used for the expressions obtained using α = 0, 1, 0.5respectively. These cases correspond for the stencils that arise using the consistent,lumped and mixed (higher-order) mass matrices respectively. The subscript e is usedto flag the choice α = αe, the optimal expression for α, in order to attain nodally ex-act numerical solutions in 1d. For the graphical representation of f(ω) and ξ(ω) wenormalize some of these fields as follows:

ω∗ :=ω

π2; ξ∗ :=

ξ

ξnq=ξ`

π(4.18)

Restricting the domain to ω∗ ∈ [0, 1] guarantees that the Nyquist frequency3 of thediscretization ((ξnq)) is always greater than the frequency of the exact solution (ξo).Thus for every wave length of the harmonic solution we ensure the presence of atleast two elements. The Nyquist-Shannon sampling theorem states that this minimumresolution of the mesh is essential to allow a perfect reconstruction of the solutionusing sinusoidal interpolation. However, using linear interpolation at least 4 elementsper wavelength (ξo` 6 (π/2) or ω∗ 6 (1/4)) are needed to capture the sinusoidalprofile. As a rule of thumb at least 8 to 10 elements per wavelength are recommendedfor a decent representation of the solution using linear interpolation [79, 183]. Thelatter resolution of the mesh is guaranteed by restricting the domain to ω∗ ∈ [0, 1/16].

Figures 58a and 58b illustrate the plot of f(ω∗) for ω∗ ∈ [0, 1] and ω∗ ∈ [0, 1/4]respectively. As expected a higher-order convergence of fm → fe is observed as ω∗ →0. Also for both the domains f(ω∗) 6 0 and in particular for the latter domain i. e.ω∗ ∈[0, 1/4], we see that |f(ω∗)| < 1. Figures 58c and 58d illustrate the plot of ξ∗(ω∗) forω∗ ∈ [0, 1] and ω∗ ∈ [0, 1/4] respectively. Whenever |f(ω∗)| > 1, Eq.(4.12) suggeststhat λ := exp(iξh`) ∈ R. This implies that ξh is a complex number (ξh ∈ C) withthe real part <(ξh) = (nπ/`),n ∈ 0, 1, 2, . . . and the imaginary part =(ξ) 6= 0. Asthe Nyquist frequency in space is ξnq = π/`, the real part is either <(ξh) = 0 forf(ω∗) > 0 or <(ξh) = (π/`) for f(ω∗) < 0. Thus whenever |f(ω∗)| > 1 we find<(ξh) = (π/`), i. e.<(ξh∗) = 1 (see Figure 58c). Also as =(ξ) 6= 0 the numericalsolutions will be subjected to amplification intrinsic to the discretization (the onestudied in the von Neumann analysis). Finally, whenever |f(ω∗)| 6 1, the solutionξh is real (ξh ∈ R) and all the considered schemes are devoid of any amplificationintrinsic to the discretization. In Figure 58d we observe that for ω∗ ∈ [0, 1/16], thegraphs of ξ∗e and ξ∗m are indistinguishable.

4.3.4 Examples

We consider the problem defined in Eq.(4.5) with the following problem data: k =

1e-3, s = 1, L = 1, Φl = 3, Φr = 1. Thus the exact solution of the problem givenby Eq.(4.6) has an angular wave number ξo = 10

√10. The discretization of the space

is done by linear finite elements and is uniform. We solve the problem using α ∈0, 1, 0.5 and the subscripts c, l,m are used to flag them respectively. Four meshes of

3 http://en.wikipedia.org/wiki/Nyquist_frequency. Here frequency is to be understood in the spatialcontext, i. e.the wavenumber


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

ω*

f(ω

* )

f(ω*) ; ω* ∈ [0,1]

fl

fm

fc

fl(ω*)

fc(ω*)

fm

(ω*)

fe(ω*)

(a)

0 1/40 2/40 3/40 4/40 5/40 6/40 7/40 8/40 9/40 1/4 −0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ω*

f(ω

* )

f(ω*) ; ω* ∈ [0,1/4]

fl(ω*)

fc(ω*)

fm

(ω*)

fe(ω*)

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

ω*

ξ*

ξ*(ω*) ; ω* ∈ [0,1]

ξ*l1

ξ*l2

ξ*m1

ξ*m2

ξ*c1

ξ*c2

ξ*l1

ξ*l2

ξ*c1

ξ*c2

ξ*m1

ξ*m2

ξ*a1

ξ*a2

(c)

0 1/40 2/40 3/40 4/40 5/40 6/40 7/40 8/40 9/40 1/4 −0.6

−0.55

−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

ω*

ξ*

ξ*(ω*) ; ω* ∈ [0,1/4]

ξ*l1

ξ*l2

ξ*c1

ξ*c2

ξ*m1

ξ*m2

ξ*a1

ξ*a2

(d)

Figure 58: Plots of f(ω∗) and ξ∗(ω∗). (a) Domain: ω∗ ∈ [0, 1] ; (b) Domain: ω∗ ∈ [0, 1/4] ; (c)Domain: ω∗ ∈ [0, 1] ; (d) Domain: ω∗ ∈ [0, 1/4]


ξo

√λm−2

√λm−1

√λm

√λm+1

√λhm−2

√λhm−1

√λhm

ξh

Figure 59: A schematic diagram that illustrates the encounter of a zone of degeneracy on meshrefinement (` → 0). As the value of

√λhm crosses ξo on its path towards

√λm, the

discrete LBB constant takes values arbitrarily close to zero.

different resolution viz. 41, 81, 162 and 323 elements are considered. These meshesguarantee the presence of at least 8, 16, 32 and 64 elements per wavelength of theharmonic solution respectively. All the meshes restrict the domain of ω∗ to [0,1/16].

Figure 60 illustrates the plots of the numerical solutions obtained using a consistent,lumped and semi-lumped mass matrices denoted by Φch, Φlh and Φmh respectively,against the exact solution of the problem denoted by Φa. In Figure 60a the solutionsΦch and Φlh are out-of-phase and as expected the phase accuracy improves on meshrefinement (Figures 60b-d). We observe a remarkable error in the amplitude of thesesolutions. Note that there is no intrinsic amplification for all the schemes and theerrors in the angular wave numbers ξ∗c, ξ∗l are small (Figure 58d). The amplitude ofthe solution depends not only on the intrinsic amplification of the scheme but alsoon the wave number ξ and on the applied Dirichlet boundary conditions. Thus wemay conclude that small errors in the wave number of the computed solution mayresult in huge errors in their amplitude. An alternative explanation to this behaviorcan be given via the following argument. First note that (

√λ10 = 31.4159) < (ξo =

10√10 = 31.6227) < (

√λ11 = 34.5575). It is possible that the discrete eigenvalue

√λh10

for the initial course mesh/grid is greater than ξo and on further mesh refinementit approaches

√λ10 by crossing ξo. This explains the observation that the numerical

solution Φch on mesh refinement first explodes (as it enters the zone of degeneracy)and then gradually converges to the exact solution. Figure 59 illustrates schematically4

the encounter of a zone of degeneracy on mesh refinement (`→ 0) while as the valueof√λhm crosses ξo on its path towards

√λm. Nevertheless, the convergence of the

discrete wavenumber (ξh → ξo) need not be affected in this process. This argumentalso suggest that this phenomenon could have been equally observed for the solutionsΦlh and Φmh should their corresponding discrete eigenvalues cross ξo.

On the other hand, the solution Φmh could represent approximately the profile ofthe exact solution even on the coarsest mesh (see Figure 60a). Figures 60b-d show thaton further mesh refinements Φmh is indistinguishable from the analytical solution.

4 A similar figure was presented earlier in [47] (c.f. Figure 1, pp. 74)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−11

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

9

10

11

x

Φh

k = 0.001 , s = 1 , f = 0 ; 41 elements ; Φl = 3 , Φr = 1

Φ

ha

Φhc

Φhl

Φhm

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−11

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

9

10

11

x

Φh


Φ

ha

Φhc

Φhl

Φhm

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−11

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

9

10

11

x

Φh


Φ

ha

Φhc

Φhl

Φhm

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−11

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

9

10

11

x

Φh


Φ

ha

Φhc

Φhl

Φhm

(d)

Figure 60: Numerical solution Φh using a mesh with at least: (a) 8 elements per wave length ;(b) 16 elements per wave length ; (c) 32 elements per wave length ; (d) 64 elementsper wave length. In figures (c) and (d) the solution Φmh effectively coincides withthe exact solution and the solutions Φcm and Φlm bound the exact solution fromabove and below respectively.


4.4 analysis in 2d

4.4.1 Introduction

In multidimensions the general solution to the problem (4.1) considering a linearsource f(x) may be expressed as follows:

φ(x) =f

s+∑θ

Cθ exp(iξθ · x) (4.19a)

|ξθ| = ξo ⇒ ξθ := (ξθ1 , ξθ2) = (ξo cos(θ), ξo sin(θ)) (4.19b)

where, Cθ represents a generic constant independent of the spatial coordinates. Gener-ally it is not possible to arrive at an expression for Cθ in the closed form. Neverthelessthis detail is not needed in the Fourier analysis of these problems. Eliminating θ fromEq.(4.19b) we arrive at the characteristic equation of the continuous problem (4.1):

(ξθ1)2 + (ξθ2)

2 = ξ2o (4.20)

4.4.2 Galerkin FEM using rectangular bilinear finite elements

The element contributions to the matrices given in Eq.(4.4) using 4-node rectangularbilinear finite elements are,

De =`26`1

2 −2 −1 1

−2 2 1 −1

−1 1 2 −2

1 −1 −2 2

+`16`2

2 1 −1 −2

1 2 −2 −1

−1 −2 2 1

−2 −1 1 2

(4.21a)

Me =`1`236

4 2 1 2

2 4 2 1

1 2 4 2

2 1 2 4

(4.21b)

where `1, `2 are the corresponding element lengths along the 2d axes. Restrainingthe discretization to be uniform, we can arrive at an equation stencil for every interiornode of the mesh. We use the following notation (described earlier in §3.1) to representa generic compact stencil obtained for the (i, j) node on a rectangular grid.

j+1, j, j−1Ai−1, i, i+1t = 0 (4.22)

where A represents the matrix of the stencil coefficients. We can guess that a solu-tion to Eq.(4.22) takes the form Φi,j := φ(xi1, xj2) = exp[i(ξh1x

i1 + ξ

h2xj2)]. Substituting

this solution into Eq.(4.22) and defining λ1 := exp(iξh1 `1) and λ2 := exp(iξh2 `2) we getthe characteristic equation of the generic stencil(4.22):

λ2, 1, λ−12

Aλ−11 , 1, λ1

t= 0 (4.23)


The stencil for the Galerkin FEM method corresponding to any interior node (i, j)can be written as Eq.(4.22) with the following definition of the stencil coefficient ma-trix (A):

Afem :=k`26`1

1, 4, 1t −1, 2,−1+k`16`2

−1, 2,−1t 1, 4, 1

−s`1`236

1, 4, 1t 1, 4, 1(4.24)

The stencil for the classical FDM method corresponding to any interior node (i, j)can be written as Eq.(4.22) with the following definition of A:

Afdm :=k`26`1

0, 6, 0t −1, 2,−1+k`16`2

−1, 2,−1t 0, 6, 0

−s`1`236

0, 6, 0t 0, 6, 0(4.25)

The characteristic equation associated with the stencil for the Galerkin FEM can bewritten as Eq.(4.23) using the definition of A given by Eq.(4.24). Likewise the charac-teristic equation associated with the stencil for the classical FDM can be written asEq.(4.23) using the definition of A given by Eq.(4.25).

4.4.3 A nonstandard compact stencil in 2d

Define two free parameters α1,α2 and consider the following definition of A:

Aα1,α2 := (1−α1)k`26`1

1, 4, 1t −1, 2,−1+α1k`26`1

0, 6, 0t −1, 2,−1

+ (1−α1)k`16`2

−1, 2,−1t 1, 4, 1+α1k`16`2

−1, 2,−1t 0, 6, 0

− (1−α2)s`1`236

1, 4, 1t 1, 4, 1−α2s`1`236

0, 6, 0t 0, 6, 0

(4.26)

Note that taking α1 = α2 = α we arrive at a stencil that is the α-interpolation of theFEM and FDM stencils, i. e.Aα,α = (1−α)Afem+αAfdm. Likewise taking α1 = 0 andα2 = α we arrive at a stencil that results from the Galerkin FEM method using an α-interpolated mass matrix Mα := (1−α)M+αML. We remark that unlike in 1d whereboth choices resulted in the same stencil, in 2d the obtained stencils are different.

Next we relate this nonstandard stencil with the compact fourth-order Padé ap-proximation in 2d. A generalized version of the same was studied in [81, 168] and theassociated stencil coefficient matrix of the scheme Aγ can be expressed as follows:

Aγ := −k

[0, 1, 0+

1,−2, 112

]t1,−2, 1`21

− k1,−2, 1t

`22

[0, 1, 0+

1,−2, 112

]

− s

[0, 1, 0+

1,−2, 112

]t [0, 1, 0+

1,−2, 112

]− s(γ− 1)

1,−2, 1t

12

1,−2, 112

(4.27)

where γ is a free parameter. The standard compact fourth-order Padé scheme in 2d isobtained by selecting γ = 1. Other alternatives viz. γ = 0 and γ = 2 were presented


in [40](cf. Appendix, Table VI, p.542). After some algebraic rearrangement, matrix Aγ

given in Eq.(4.27) can be re-written equivalently as follows:

Aγ :=k

2

[1, 4, 16

+0, 6, 06

]t−1, 2,−1

`21+k

2

−1, 2,−1t

`22

[1, 4, 16

+0, 6, 06

]

−s

2

[1, 4, 1t

6

1, 4, 16

+0, 6, 0t

6

0, 6, 06

]− s(γ− 2)

1,−2, 1t

12

1,−2, 112

(4.28)

Note that by selecting γ = 2we obtain a stencil that is equivalent to the one obtainedby taking the average of the FEM and the FDM stencils. Thus,

A2 =1

`1`2A0.5,0.5 (4.29)

We now relate this nonstandard stencil for square meshes with the compact schemeproposed by Vichnevetsky and Bowles [187] in order to reduce the anisotropy relatedto the numerical dispersion. This scheme was studied in [183] and the conditionsfor appropriate numerical isotropy were determined therein. Also, this scheme wasused to synthesize an equivalent transmission-line matrix (TLM) [115] model for theMaxwell’s equations in [167]. The associated stencil coefficient matrix Avb can bewritten as follows.

Avb :=γk

`2

0 −1 0

−1 4 −1

0 −1 0

+

(1− γ)k

2`2

−1 0 −1

0 4 0

−1 0 −1

− s

0 0 0

0 1 0

0 0 0

(4.30)

where γ is the associated interpolation parameter. Note that for γ = 1 we recoverthe classical FDM (i. e.the second-order central difference scheme) and for γ = 0 weget a similar scheme but with the stencil inclined at 45 and hence with the meshsize

√2`. Note that we recover the Galerkin FEM contribution of the term −k∆φ by

choosing γ = (1/3). Making the substitution γ = (1 + 2α)/3 in Eq.(4.30) and aftersome algebraic rearrangement, matrix Avb can be re-written equivalently as follows:

Avb := k

[(1−α)

61, 4, 1+

α

60, 6, 0

]t−1, 2,−1

`2

+ k−1, 2,−1t

`2

[(1−α)

61, 4, 1+

α

60, 6, 0

]− s

0, 6, 0t

6

0, 6, 06

(4.31)

This is precisely what we get using an α-interpolated (Galerkin FEM and classicalFDM) diffusion matrix in the classical FDM stencil. Thus, on square meshes we canrelate Avb with the nonstandard stencil as shown below.

Avb =1

`2Aα,1 (4.32)


Using the definition of A given by Eq.(4.26), the characteristic equation associated tothe resulting stencil is given by Eq.(4.33) and on simplification we arrive at Eq.(4.34).

λ2, 1, λ−12

Aα1,α2λ−11 , 1, λ1

t= 0 (4.33)

⇒([(1−α1)(λ

22 + 4λ2 + 1) + 6α1λ2](−1+ 2λ1 − λ

21)

6ω1

)

+

((−λ22 + 2λ2 − 1)[(1−α1)(1+ 4λ1 + λ

21) + 6α1λ1]

6ω2

)

−

([(1−α2)(λ

22 + 4λ2 + 1)(1+ 4λ1 + λ

21) + 36α2λ2λ1]

36

)= 0

(4.34)

where ω1,ω2 are two dimensionless element numbers defined as follows:

ω1 :=s`21k

= (ξo`1)2 ; ω2 :=

s`22k

= (ξo`2)2 (4.35)

Unlike in 1d, the characteristic equations of the stencils in 2d have infinite solutions(fundamental frequencies (ξh1 , ξh2 )) for every (ω1,ω2) pair. For every choice of thepair (ω1,ω2), these solutions will trace well-defined contours in the ξh1 − ξh2 plane.The solutions to Eq.(4.34) are symmetric about the origin and the axes. This statementcan be easily verified due to the fact that by replacing the pair (λ1, λ2) with (λ±11 , λ±12 )

in Eq.(4.34) we end up in the same equation. Thus we may conclude that if (ξh1 , ξh2 )is a solution to Eq.(4.34) then (±ξh1 ,±ξh2 ) are also solutions to the same. Obviouslythis statement also extends to the characteristic equation of the continuous problem(4.20) which additionally has a rotational symmetry (i. e.if (ξθ1 , ξθ2) is a solution then(ξθ2 , ξθ1) is also a solution). These contour lines are circular for the continuous problemand their radius equals to the chosen ξo value. Rotational symmetry for the solution(ξh1 , ξh2 ) is attained should the element lengths be the same, i. e.`1 = `2 = `. In thiscase the stencil coefficient matrix Aα1,α2 is symmetric and after scaling down by k itcan be expressed as follows:

Aα1,α2

k=

A2 A1 A2

A1 A0 A1

A2 A1 A2

α1,α2

;

Aα1,α20 :=

8

3−4ω

9+4α13

−5ωα29

Aα1,α21 := −

1

3−ω

9−2α13

+ωα29

Aα1,α22 := −

1

3−ω

36+α13

+ωα236

(4.36)

where, ω := (s`2/k) = (ξo`)2.

Finally we relate this nonstandard stencil with methods that have a symmetric sten-cil coefficient matrix Asym defined as:

Asym :=

A2 A1 A2

A1 A0 A1

A2 A1 A2

(4.37)

Let g1 = (4A1/A0) and g2 = (4A2/A0) and if (g1 + g2 + 1) 6= 0 then we can obtainAsym (possibly scaled by a factor) from Aα1,α2 by selecting α1 and α2 as follows:

α1 :=4(g1 + g2 + 1) +ω(g1 − 4g2)

8(g1 + g2 + 1); α2 :=

12(g1 + g2 + 1) +ω(2− g1 − 4g2)

2ω(g1 + g2 + 1)


(4.38)

For instance consider the QSFEM method [10] for which the expressions for g1 andg2 can be written as shown below.

g1 :=2(c1s1 − c2s2)

c2s2(c1 + s1) − c1s1(c2 + s2); g2 :=

(c2 + s2 − c1 − s1)

c2s2(c1 + s1) − c1s1(c2 + s2)

(4.39a)

c1 := cos[√ω cos

( π16

)]c2 := cos

[√ω cos

(3π

16

)]

s1 := cos[√ω sin

( π16

)]s2 := cos

[√ω sin

(3π

16

)] (4.39b)

4.4.4 Numerical solution, phase error and local truncation error

In this section we will deal only with the case when `1 = `2 = `. Here we presentthe solution to Eq.(4.34) for a given α1,α2 expressed as a generic series expansion interms of ω as follows:

α1 :=

∞∑m=0

amωm ≈ a0 + a1ω+ a2ω

2 + a3ω3 +O

(ω4)

(4.40a)

α2 :=

∞∑m=0

bmωm ≈ b0 + b1ω+ b2ω

2 + b3ω3 +O

(ω4)

(4.40b)

where am,bm are coefficients independent of ω. Following [10] the solution ξh :=

(ξh1 , ξh2 ) can also be expressed as a series expansion in terms of ω:ξh1 `

ξh2 `

= R(am,bm,β,ω)

cos(β)

sin(β)

(4.41a)

R :=√ω

[1+

∞∑m=1

rm(ai,bi,β)ωm]≈√ω[1+ r1ω+ r2ω

2 + r3ω3 +O

(ω4)]

(4.41b)

where, rm are coefficients independent of ω and will be determined later in thissection. Recall that the numerical solution in 1d given by Eq.(4.13) or Eq.(4.14) obeysthe above series expansion in terms of ω. Figure 61 illustrates schematically the con-tour traced by the numerical solution Ph(ξh1 `, ξ

h2 `) and compares it with the contour

of the exact solution P(ξβ1 `, ξβ2 `). In [10] the denomination ‘dist(β)’ was used for the

distance between Ph and P, i. e.dist(β) := R −√ω. Therein ‘dist(β)’ was used as a

measure of the approximation quality of the solution and from it error estimates werederived that bound the solution from below. The relative phase error of the solutionalong any direction β is given by,

‖ξh‖− ‖ξβ‖‖ξβ‖

=R−√ω√

ω=

dist(β)√ω

=

∞∑m=1

rm ωm ≈

[r1ω+ r2ω

2 + r3ω3 +O

(ω4)]

(4.42)


β

PhP

O

P :=√ω(cosβ, sinβ)

Ph := R(β, ω)(cos β, sinβ)

R(β, ω) :=√ω

[1 +

∞∑

m=1

rm(β)ωm

]

ξh1 ℓ

ξh2 ℓ

Figure 61: A schematic diagram of the contours traced by the numerical solution Ph(ξh1 `, ξh2 `)

and the exact solution P(ξβ1 `, ξβ2 `).

Substituting Ph(ξh1 `, ξh2 `) into the stencil corresponding to Aα1,α2 given in Eq.(4.36)

we get:

Aα1,α20 + 2Aα1,α2

1 [cos(R cosβ)+ cos(R sinβ)] + 4Aα1,α22 cos(R cosβ) cos(R sinβ) = 0

(4.43)

Using the definitions of α1,α2 and R given in Eq.(4.40) and Eq.(4.41b) respectively,the left hand side (LHS) of Eq.(4.43) can be expanded as a series in terms of ω asshown in Eq.(4.44a). The first four coefficients of this series can be expressed as shownin Eq.(4.44b) and Eq.(4.44c) respectively.

LHS =

∞∑m=0

Sm(ai,bj, rk,β)ωm (4.44a)

S0 = S1 = 0 ; S2 = 2r1 +

(3+ 2a0 − 8b0

48

)+

(1− 2a048

)cos(4β) (4.44b)

S3 = 2r2 + r21 +

(2a0 − 4b0 − 1

12

)r1 +

(24a1 − 96b1 − 2a0 + 8b0 − 5

576

)

+

[(10a0 − 7− 120a1

2880

)+

(1− 2a012

)r1

]cos(4β)

(4.44c)

The local truncation error of the solution along any direction β is found by substi-tuting the exact solution P(ξβ1 `, ξ

β2 `) into the stencil corresponding to Aα1,α2 given in

Eq.(4.36). This is equivalent to substituting rk = 0 ∀ k in the expression for LHS givenin Eq.(4.44a). Thus using the result S0 = S1 = 0, the relative truncation error T alongany direction β is given by:

T :=LHS|Pω

=

∞∑2

Sm(ai,bj, rk = 0,β)ωm−1 ≈[S2ω+ S3ω

2 + S4ω3 +O

(ω4)]

(4.45)

We now present the expressions for the unknowns rk. Clearly all the coefficients Smshould be zero for Eq.(4.43) to hold. We can solve for the unknowns rk by imposing


the conditions Sm = 0 ∀ m. Thus the first two unknowns in Eq.(4.41b) viz. r1 and r2can be expressed as follows:

r1 = −

(3+ 2a0 − 8b0

96

)−

(1− 2a096

)cos(4β) (4.46a)

r2 = −r212

−

(2a0 − 4b0 − 1

24

)r1 −

(24a1 − 96b1 − 2a0 + 8b0 − 5

1152

)

−

[(10a0 − 7− 120a1

5760

)+

(1− 2a024

)r1

]cos(4β)

(4.46b)

Note that we obtain the condition r1 = 0 if and only if a0 and b0 satisfy thecondition a0 = b0 = (1/2). Further we obtain the condition r2 = 0 if and only if a1and b1 satisfy the condition a1 = (−1/60) and b1 = (−1/40). For these choices ofa0,a1,b0 and b1 the first five coefficients in Sm can be simplified as follows:

S0 = S1 = 0 ; S2 = 2r1 ; S3 = 2r2 + r21 −

r16

(4.47a)

S4 = 2r3 + 2r1r2 −r26

−r1720

−5r2112

−

[5

55296−

(a2 − 4b224

)+

(1+ 576a213824

)cos(4β) +

cos(8β)387072

] (4.47b)

Likewise, by imposing the condition S4 = 0 the unknown r3 in Eq.(4.41b) can besimplified to the following:

r3 =

[5

110592−

(a2 − 4b248

)+

(1+ 576a227648

)cos(4β) +

cos(8β)774144

](4.48)

Clearly it is impossible to obtain the condition r3 = 0 and this fact was pointedout earlier in [10]. To conclude this section we summarize the salient results. Theparameters α1 and α2 that appear in Aα1,α2 can be chosen such that the numericalsolution be sixth-order accurate, i. e.O

((ξo`)

6)

or equivalentlyO(ω3). Recall that this

is the maximum order of accuracy that can be attained on any compact stencil [10].All such α1 and α2 should obey the following series expansion in terms of ω.

α1 =1

2−ω

60+

∞∑m=2

amωm ; α2 =

1

2−ω

40+

∞∑m=2

bmωm (4.49)

The relative phase and local truncation errors of these schemes can be expressed asfollows:


= r3ω3 +O

(ω4)

; T = −2r3ω3 +O

(ω4)

(4.50)

where r3 is given in Eq.(4.48). As am,bm (m > 2) can be chosen arbitrarily, infinitelymany sixth-order schemes can be designed through Aα1,α2 . Of course some particularchoice of am,bm may yield a scheme with better features. For instance, am,bm maybe chosen such that the local truncation error T be zero along some chosen directions.


4.4.5 α-Interpolation of the FEM and the FDM in 2d

In this section we consider the case α1 = α2 = α, that results in a scheme which is theα-interpolation of the FEM and the FDM stencils. Here the coefficients am = bm ∀ m.Recall that a necessary condition to obtain a sixth-order scheme is a1 = (−1/60) andb1 = (−1/40). Thus an immediate consequence is that this α-interpolation schemecan be at the best fourth-order accurate. Nevertheless, a compromise to the loss inaccuracy is that the condition α1 = α2 = α imposes an additional structure to thescheme that may be exploited. For instance, this additional structure might throwlight on the extension of this scheme to unstructured meshes. Precisely, should it bepossible to design a Petrov–Galerkin method that would yield the FDM stencil ona structured mesh, then this scheme can be extended to unstructured meshes in astraight-forward manner. We show that indeed it is possible to design such a Petrov–Galerkin method using just the lowest-order block finite elements in chapter 5 and[138].

We now discuss the salient features of this scheme, i. e.the case α1 = α2 = α. It ispossible to choose α such that the local truncation error along any direction θ be zero.Let this choice be denominated as αθ and it can be expressed as follows:

αθ :=6(cθ + sθ + 2cθsθ − 4) +ω(2cθ + 2sθ + cθsθ + 4)

12(1− cθ − sθ + cθsθ) +ω(2cθ + 2sθ + cθsθ − 5);

cθ := cos(√ω cos(θ))

sθ := cos(√ω sin(θ))

(4.51)

Note that choosing θ = 0 we would recover the expression for α given in Eq.(4.16)which results in solutions that are nodally exact in 1d. The expression for αθ can bewritten as a series expansion in terms of ω as shown below:

αθ =

∞∑m=0

amωm ≈ 1

2−

[5+ cos(4θ)3+ cos(4θ)

]ω

60−

[35+ 28 cos(4θ) + cos(8θ)

3+ cos(4θ)

]ω2

16128

+O(ω3)

(4.52)

Recall that the choice a0 = (1/2) will make the coefficient r1 = 0 and hence us-ing the expression for αθ we will always obtain fourth-order accurate solutions onuniform meshes. The expression for the coefficient r2 given in Eq.(4.46b) can now besimplified to the following.

r2 =1

1440

[cos(4θ) − cos(4β)3+ cos(4θ)

](4.53)

The relative phase and local truncation errors of this scheme can be expressed asfollows:


= r2ω2 +O

(ω3)

; T = −2r2ω2 +O

(ω3)

(4.54)

where r2 is given in Eq.(4.53). So far the direction θ, along which the local truncationerror is made zero, is arbitrary. We now try to optimize the solution error with re-spect to θ. Ideally the function to optimize could be either the relative phase or localtruncation errors and the optimization problem can be posed as follows:

minθ

maxβ

|T| (or) minθ

maxβ

∣∣∣∣∣‖ξh‖− ‖ξβ‖‖ξβ‖

∣∣∣∣∣ (4.55)


Unfortunately, this is a difficult problem to solve in the closed form as it is a nonlin-ear function ofω and the location of the minimum might vary withω. We conjuncturethat in the pre-asymptotic range (i. e.ξo` 1 or equivalently ω 1) the location ofthe minimum in the θ−β space is independent ofω. Thus, under this assumption theminimization of the relative phase or local truncation errors is essentially equivalentto the minimization of the coefficient of the lowest order term, i. e.here r2. Hence wechoose to optimize the coefficient r2 instead. The redefined problem and its solutionis given below.

minθ

maxβ

|r2| = minθ

maxβ

| cos(4θ) − cos(4β)|1440(3+ cos(4θ))

= minθ

1+ | cos(4θ)|1440(3+ cos(4θ))

=1

4320

(4.56a)

maxβ

occurs at | cos(4β)| = 1 ⇒ β =mπ

4; m = 0, 1, 2, . . .

minθ

occurs at | cos(4θ)| = 0 ⇒ θ =(2n+ 1)π

8; n = 0, 1, 2, . . .

(4.56b)

Thus, for a given θ the maximum error in the stencil will be found for some β ∈0, (π/4), (π/2). That maximum error along the direction β takes a minimum valueshould the chosen direction (where the truncation error is made zero) be some θ ∈(π/8, 3π/8). Note that due to the inherent symmetries in the stencil the expressionsfor α(π/8) and α(3π/8) are equivalent.

4.4.6 Dispersion plots in 2d

For a feasible graphical representation and comparison of the solutions to the char-acteristic equations we plot the ξ1 − ξ2 contours for some values of (ω1,ω2) only.Here and henceforth the superscripts θ,h are dropped in order to refer to the con-tour plots of both the continuous and discrete problems simultaneously. In order toretain generality to the plots the quantities ω1,ω2, ξθ1 , ξθ2 , ξh1 and ξh2 are normalizedas follows:

ω∗1 :=ω1π2

; ξθ∗1 :=ξθ1ξnq1

=ξθ1`1

π; ξh∗1 :=

ξh1ξnq1

=ξh1 `1

π(4.57a)

ω∗2 :=ω2π2

; ξθ∗2 :=ξθ2ξnq2

=ξθ2`2

π; ξh∗2 :=

ξh2ξnq2

=ξh2 `2

π(4.57b)

⇒ λ1 := eiξh1 `1 = eiπξ

h∗1 ; λ2 := e

iξh2 `2 = eiπξh∗2 (4.57c)

where ξnq1 , ξnq2 are the Nyquist frequencies of the discretization along the 2d axes.Using these normalized quantities the characteristic equations of the continuous and


discrete problems given by Eq.(4.20) and Eq.(4.34) can be expressed as Eq.(4.58) andEq.(4.59) respectively.

(ξθ∗1 )2

ω∗1+

(ξθ∗2 )2

ω∗2= 1 (4.58)

([(1−α1)(λ

22 + 4λ2 + 1) + 6α1λ2](−1+ 2λ1 − λ

21)

6π2ω∗1

)

+

((−λ22 + 2λ2 − 1)[(1−α1)(1+ 4λ1 + λ

21) + 6α1λ1]

6π2ω∗2

)

−

([(1−α2)(λ

22 + 4λ2 + 1)(1+ 4λ1 + λ

21) + 36α2λ2λ1]

36

)= 0

(4.59)

For every choice of the pair (ω∗1,ω∗2) the solution to Eq.(4.58) will trace ellipticcontours with the center at the origin in the ξ∗1 − ξ

∗2 plane. Due to the inherent sym-

metry of the solutions the dispersion plots are presented just in the first quadrant.Similar to 1d, we require that the Nyquist frequencies of the discretization in 2d arealways greater than the frequencies of the exact solution, i. e.minξnq1 , ξnq2 > ξo.Note that the following expressions are equivalent (≡): minξnq1 , ξnq2 ≡ max`1, `2 ≡maxω∗1,ω∗2. Thus restricting the domain to maxω∗1,ω∗2 ∈ [0, 1] guarantees this re-quirement :

ω∗ := maxω∗1,ω∗2 ∈ [0, 1]⇔ minξnq1 , ξnq2 > ξo (4.60)

Likewise, a mesh resolution of at least 8 elements per wavelength is guaranteed byrestricting the domain to ω∗ ∈ [0, 1/16]. We study the following four cases concernedwith the choice of the (α1,α2) pair:

I: α1 = α2 = (1/2). This case corresponds to the equation stencil associated withA0.5,0.5 := (Afem + Afdm)/2. Thus the discrete system obtained here is the av-erage of the systems obtained from the Galerkin FEM and the classical FDM.Recall that we can also obtain this stencil using the generalized Padé approxi-mation in 2d and choosing the parameter γ = 2.

II: α1 = α2 = αθ and θ = 0. This case corresponds to the α-interpolation of theGalerkin FEM and the classical FDM. The local truncation error is zero alongthe direction θ = 0 whenever `1 = `2.

III: α1 = α2 = αθ and θ = (π/8). This case also corresponds to the α-interpolationof the Galerkin FEM and the classical FDM. The local truncation error is zeroalong the direction θ = (π/8) whenever `1 = `2. Recall that choosing θ = (π/8)

leads to an optimized expression for the coefficient r2.

IV: QSFEM, α1 6= α2 6= 0 and given by Eq.(4.38) and Eq.(4.39). This case correspondsto the quasi-stabilized FEM presented in [10]. The local truncation error is zeroalong the directions θ = (π/16) and θ = (3π/16) whenever `1 = `2.

Note that for cases I,II and III the relative phase and local truncation errors ofthe numerical solution diminish at a fourth-order rate i.e O

((ξo`)

4)

or equivalentlyO(ω2). For the case IV, i.e the QSFEM, these errors diminish at a sixth-order rate i.e

O((ξo`)

6)

or equivalently O(ω3).


In Figures 62 and 63 we plot the solutions to the characteristic equations of thecontinuous and discrete problems given by Eq.(4.58) and Eq.(4.59) respectively. Thecontours of the continuous problem are drawn using the dashed line-style and thecorresponding contour value displayed in a single text-box. Labeled solid line-style isused to display the contours of the discrete problem. Each figure is further dividedinto four sub-figures viz. (a)-(d) which correspond to the considered four cases I–IV.Within each sub-figure the contours plots of the continuous and discrete problems areplotted and compared. In Figure 62 we plot the ξ∗1 − ξ

∗2 contours keeping ω∗1 = ω∗2

i. e.`1 = `2. The plotting domain considered here is (ξ∗1, ξ∗2) = [0, 0.55]× [0, 0.55]. InFigure 63 we plot the ξ∗1 − ξ

∗2 contours keeping ω∗2 = 0.49ω∗1 i. e.`2 = 0.7`2. The

plotting domain considered here is again (ξ∗1, ξ∗2) = [0, 0.55] × [0, 0.55]. In both thefigures, contours are drawn for the values of ω∗ ∈ (1/4), (1/9), (1/16), (1/25). Thesevalues of ω∗ guarantee the presence of at least four, six, eight and ten elements perwavelength respectively. Note that except for the contour value ω∗ = (1/4) in caseI, the rest of the contours of the numerical solution are indistinguishable from theircontinuous counterparts. This is due to the fact that the relative local truncation erroris of the order of 1e-3 which is small with respect to the scale of the plotting domain.

In order to quantify better the relative local truncation errors of the solutions, wecompare them in Figure 64. This figure is further divided into four sub-figures viz.(a)-(d) which correspond to the considered four values of ω∗ respectively, i. e.ω∗ ∈(1/4), (1/9), (1/16), (1/25). Within each sub-figure the relative local truncation errorsof the considered four cases viz. I–IV are plotted vs. the direction β. Now we canclearly distinguish the errors related to the four cases. However in these figures theerror associated with the case IV, i. e.the QSFEM is indistinguishable from zero at thisscale. In Figure 65 the relative local truncation errors of the solutions are plotted inthe log-scale. The sub-figures are organized just like in Figure 64. Note that in figures64 and 65 the relative local truncation errors converge monotonically with respect toω∗, i. e.the plots of the errors with respect to the direction β maintain their shape.This supports the conjuncture made in Section 4.4.5 that in the pre-asymptotic rangethe location of the mini-max error is independent of ω∗. Also we note that choosingθ = (π/8) in the expression for αθ, the maximum error is less than the one choosingθ = 0.

4.4.7 Examples

We consider the problem defined by Eq.(4.1) with the following problem data: k ∈1e-3, 1e-4, s = 1, f = 0 and the domain Ω = [0, 1]× [0, 1]. The Dirichlet boundaryconditions are assigned such that the exact solution of Eq.(4.1) is φ(x) = sin(ξβ · x),where β is the chosen direction of wave propagation, ξβ := ξo(cos(β), sin(β)) andξo :=

√s/k. Thus for the chosen values of k, the wavenumber ξo takes the values in

10√10, 100. The following wave directions are considered: β ∈ (π/9), (π/4). Seven

uniform meshes (`1 = `2) of different resolution are considered such that there areat least four, six, eight, ten, twelve, fourteen and sixteen elements per wavelengthrespectively. If the element length is chosen such that there are exactly n elements perwavelength, then the value of ξ∗ = (2/n) and ω∗ = (2/n)2. As it can be seen all these


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.250.25

1/4

0.11111

0.11111

0.11111

0.11111

0.11111

0.11111

1/9

0.0625

0.0625

0.0625

0.0625

1/16

0.04

0.04

0.04

0.04

1/25

ξ*1

ξ* 2

ξ*1−ξ*

2 contours ; α

1 = α

2 = 0.5

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.250.25

0.25

0.25

0.25

0.25

0.25

0.250.25

1/4

0.11111

0.11111

0.11111

0.11111

0.11111

0.11111

1/9

0.0625

0.0625

0.0625

0.0625

1/16

0.04

0.04

0.04

0.04

1/25

ξ*1

ξ* 2

ξ*1−ξ*

2 contours ; α

1 = α

2 = α

θ ; θ = 0

(b)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.250.25

1/4

0.11111

0.11111

0.11111

0.11111

0.11111

0.11111

1/9

0.0625

0.0625

0.0625

0.0625

1/16

0.04

0.04

0.04

1/25

ξ*1

ξ* 2

ξ*1−ξ*

2 contours ; α

1 = α

2 = α

θ ; θ = π/8

(c)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.250.25

0.25

0.25

0.25

0.25

0.25

0.250.25

1/4

0.11111

0.11111

0.11111

0.11111

0.11111

0.11111

1/9

0.0625

0.0625

0.0625

0.0625

1/16

0.04

0.04

0.04

0.04

1/25

ξ*1

ξ* 2

ξ*1−ξ*

2 contours ; α

1 ≠ α

2 ; QSFEM

(d)

Figure 62: ξ∗1 − ξ∗2 contours for ω∗ ∈ (1/4), (1/9), (1/16), (1/25) and ω∗1 = ω∗2. The dashed

and solid line-styles correspond to the solutions of the continuous and discreteproblems respectively. (a) Case I: α1 = α2 = 0.5 ; (b) Case II: α1 = α2 = αθ andθ = 0 ; (c) Case III: α1 = α2 = αθ and θ = (π/8) ; (d) Case IV: QSFEM, α1 6= α2 6= 0and given by Eq.(4.38) and Eq.(4.39)


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.25

0.25

0.25

0.25

0.25

0.25

0.250.25

1/4

0.11111

0.11111

0.11111

0.11111

0.11111

1/9

0.0625

0.0625

0.0625

0.0625

1/16

0.04

0.04

0.04

1/25

ξ*1

ξ* 2

ξ*1−ξ*

2 contours ; α

1 = α

2 = 0.5

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.25

0.25

0.25

0.25

0.25

0.25

0.250.25

1/4

0.11111

0.11111

0.11111

0.11111

0.11111

1/9

0.0625

0.0625

0.0625

1/16

0.04

0.04

0.04

1/25

ξ*1

ξ* 2

ξ*1−ξ*

2 contours ; α

1 = α

2 = α

θ ; θ = 0

(b)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

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0.25

0.25

0.25

0.25

0.25

0.25

0.250.25

1/4

0.11111

0.11111

0.11111

0.11111

0.11111

1/9

0.0625

0.0625

0.0625

0.0625

1/16

0.04

0.04

0.04

1/25

ξ*1

ξ* 2

ξ*1−ξ*

2 contours ; α

1 = α

2 = α

θ ; θ = π/8

(c)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.25

0.25

0.25

0.25

0.25

0.25

0.250.25

1/4

0.11111

0.11111

0.11111

0.11111

0.11111

1/9

0.0625

0.0625

0.0625

1/16

0.04

0.04

0.04

1/25

ξ*1

ξ* 2

ξ*1−ξ*

2 contours ; α

1 ≠ α

2 ; QSFEM

(d)

Figure 63: ξ∗1−ξ∗2 contours forω∗ ∈ (1/4), (1/9), (1/16), (1/25) andω∗2 = 0.49ω∗1. The dashed

and solid line-styles correspond to the solutions of the continuous and discreteproblems respectively. (a) Case I: α1 = α2 = 0.5 ; (b) Case II: α1 = α2 = αθ andθ = 0 ; (c) Case III: α1 = α2 = αθ and θ = (π/8) ; (d) Case IV: QSFEM, α1 6= α2 6= 0and given by Eq.(4.38) and Eq.(4.39)


0 Pi/16 Pi/8 3Pi/16 Pi/4 5Pi/16 3Pi/8 7Pi/16 Pi/2−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

β

Rel

ativ

e LT

E

Relative local truncation error (LTE) for ω* = 1/4

α1=α

2 = 0.5

α1=α

2 = α

θ; θ = 0

α1=α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

(a)

0 Pi/16 Pi/8 3Pi/16 Pi/4 5Pi/16 3Pi/8 7Pi/16 Pi/2−5

−4

−3

−2

−1

0

1x 10

−3

β

Rel

ativ

e LT

E


α1=α

2 = 0.5

α1=α

2 = α

θ; θ = 0

α1=α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

(b)


−14

−12

−10

−8

−6

−4

−2

0

2x 10

−4

β

Rel

ativ

e LT

E


α1=α

2 = 0.5

α1=α

2 = α

θ; θ = 0

α1=α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

(c)


−6

−5

−4

−3

−2

−1

0

1

2x 10

−4

β

Rel

ativ

e LT

E


α1=α

2 = 0.5

α1=α

2 = α

θ; θ = 0

α1=α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

(d)

Figure 64: Relative local truncation error plots using `1 = `2. Comparisons are made for theconsidered four cases viz. I–IV and for (a) ω∗ = (1/4) ; (b) ω∗ = (1/9) ; (c) ω∗ =

(1/16) ; (d) ω∗ = (1/25)


0 Pi/16 Pi/8 3Pi/16 Pi/4 5Pi/16 3Pi/8 7Pi/16 Pi/210

−6

10−5

10−4

10−3

10−2

10−1

β

|Rel

ativ

e LT

E|


α1=α

2 = 0.5

α1=α

2 = α

θ; θ = 0

α1=α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

(a)


−7

10−6

10−5

10−4

10−3

10−2

β

|Rel

ativ

e LT

E|


α1=α

2 = 0.5

α1=α

2 = α

θ; θ = 0

α1=α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

(b)


−7

10−6

10−5

10−4

10−3

10−2

β

|Rel

ativ

e LT

E|


α1=α

2 = 0.5

α1=α

2 = α

θ; θ = 0

α1=α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

(c)


−8

10−7

10−6

10−5

10−4

10−3

β

|Rel

ativ

e LT

E|


α1=α

2 = 0.5

α1=α

2 = α

θ; θ = 0

α1=α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

(d)

Figure 65: Log-scaled relative local truncation error plots using `1 = `2. Comparisons aremade for the considered four cases viz. I–IV and for (a) ω∗ = (1/4) ; (b) ω∗ = (1/9)

; (c) ω∗ = (1/16) ; (d) ω∗ = (1/25)


meshes restrict the domain of ω∗ to [0, 1/4]. For these considerations we study theconvergence of the relative error in the following error norms:

L2 norm‖φ−φh‖0‖φ‖0

:=[∫Ω(φ−φh)

2 dΩ]1/2

[∫Ωφ

2 dΩ]1/2(4.61a)

H1 semi-norm‖φ−φh‖1‖φ‖1

:=[∫Ω |∇(φ−φh)|

2 dΩ]1/2

[∫Ω |∇φ|2 dΩ]1/2

(4.61b)

l∞ Euclidean norm|Φe −Φh|∞

|Φe|∞ :=maxi |Φie −Φih|

maxi |Φie|(4.61c)

In the convergence studies done here, the numerical solutions corresponding tothe four cases viz. I–IV, are compared with the following solutions: the nodally exactinterpolant denoted by Ihφ and the best approximations with respect to the L2 normand the H1 semi-norm denoted by P0hφ and P1hφ respectively. The solutions Ihφ, P0hφand P1hφ can be found as shown in Eq.(4.62).

Ihφ := NaΦae (4.62a)∫Ω

wh(φ− P0hφ) dΩ = 0 ∀ wh ∈ Vh0⇒ ‖ φ− P0hφ ‖0 6 ‖ φ−φh ‖0 ∀ φh ∈ Vh

(4.62b)

∫Ω

∇wh ·∇(φ− P1hφ) dΩ = 0 ∀ wh ∈ Vh0⇒ ‖ φ− P1hφ ‖1 6 ‖ φ−φh ‖1 ∀ φh ∈ Vh

(4.62c)

As here the exact solution is sinusoidal, we have used a third-order Gauss quadra-ture rule to evaluate the expressions in Eq.(4.61) and Eq.(4.62). Figures 66a and 66billustrate the convergence of the relative error considering the wavenumber ξo =

10√10 ≈ 31.62, the wave direction β = (π/9) and using the L2 norm and H1 semi-

norm respectively. Figures 66c and 66d illustrate the same but now considering thewave direction β = (π/4). Clearly the errors in the L2 norm and H1 semi-norm corre-sponding to all the cases are greater than that of the respective best approximations.The error lines corresponding to the cases II–IV show a convergence trend indistin-guishable from the error line of Ihφ. On coarse meshes the error line correspondingto case I deviates significantly from the error line of Ihφ. Nevertheless, it quicklyrecovers the convergence trend of the later on finer meshes.

Figures 67a and 67b illustrate the convergence of the relative error considering thewavenumber ξo = 100, the wave direction β = (π/9) and using the L2-norm andH1-seminorm respectively. Figures 67c and 67d illustrate the same but now consid-ering the wave direction β = (π/4). Note that a higher value of ξo does introducethe ‘pollution-effect’ in the error lines as they deviate more from the error line of Ihφ.However the pollution effect is very small for cases II and III and is practically nil forcase IV (sixth-order dispersion accuracy).

Figure 68 illustrates the convergence of the relative error in the l∞ Euclidean norm.As a nodally exact solution requires that the dispersion error be zero, one may ex-pect that the order of convergence in the l∞ Euclidean norm be the same as that ofthe corresponding dispersion error. In fact the same is observed for the solutions cor-responding to all the cases. The error lines of cases I–III converge at a fourth-orderrate and that of case IV converges at a sixth-order rate. The error lines of the best


approximations in the L2 norm (P0hφ) and the H1 semi-norm (P1hφ) converge at asecond-order rate. The relative error of P0hφ is always greater that that of P1hφ. Thepollution effect is now clearly visible for all the cases. Irrespective of the wave direc-tion β, the error lines of all the cases shift higher with an increase in the wavenumberξo. Meanwhile, the location of the error lines of P0hφ and P1hφ are practically unaf-fected by an increase in ξo (no pollution). As the magnitudes of the relative error inthe l∞ Euclidean norm for cases II–IV is small (with respect to relative error of Ihφ inthe the L2 norm and the H1 semi-norm) for both the values of ξo, the pollution effectis hardly visible for these cases considering the relative error in the L2 norm and theH1 semi-norm.

Remark: As discussed in Section 4.3.4 and pointed out earlier in [47], though thediscrete LBB constant in an average sense is inversely proportional to ξo, it has amore complicated behavior that tends its value to zero should ξo approach the zonesof degeneracy (see Figure 59). Thus pollution effects may be found not only for higherwavenumbers but also in those situations where the wavenumber ξo approaches thezones of degeneracy. Of course, the higher the dispersion accuracy the closer will bethe discrete eigenvalues to their continuous counterparts and narrower will be thezones of degeneracy. Also, if only the Dirichlet boundary conditions are prescribed(as is the case here), spurious amplitude and/or phase modulations might occur tosatisfy them in spite of small dispersion errors [79]. For the presented scheme, wehave found vestiges of this behavior along the wave direction β = 0.

4.5 conclusions and outlook

A fourth-order compact scheme on structured meshes is presented for the Helmholtzequation. The scheme consists in taking the α-interpolation of the Galerkin FEM andthe classical FDM. For the 2d analysis of this scheme a generic nonstandard compactstencil involving two parameters α1,α2 is considered. In particular this nonstandardcompact stencil can model the aforementioned scheme (choosing α1 = α2 = α) andalso the QSFEM which has a dispersion accuracy of sixth-order. The expression forthe numerical solution of this nonstandard stencil is given considering generic ex-pressions for α1,α2 written as a series expansion in terms of ω := (ξo`)

2. Using thisresult, we provide the expressions for the phase and local truncation errors of thisnonstandard compact stencil. In particular for our scheme it is shown that these er-rors diminish at the rate O

((ξo`)

4)

or equivalently O(ω2). An expression for the

parameter α is given that minimizes the relative phase error in the pre-asymptoticrange (ξo` small). Also, by this choice the local truncation error of the scheme alongthe direction β = (π/8) is made zero. Convergence studies of the relative error in theL2 norm, the H1 semi-norm and the l∞ Euclidean norm are done and the pollutioneffect is found to be small. In particular, using the optimal expression for α the rela-tive error of our scheme in the l∞ Euclidean norm (for the considered examples andusing at least ten elements per wavelength) is found to be around or less than onepercent.

The abstractness in the definition of the QSFEM hinders its extension to unstruc-tured meshes. This is a common problem faced by all the sixth-order methods pro-posed within the framework of the FDM. The recently proposed QOPG method ad-dresses this issue and is able to attain a dispersion accuracy of the same order as


−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2−2.5−2.4−2.3−2.2−2.1

−2−1.9−1.8−1.7−1.6−1.5−1.4−1.3−1.2−1.1

−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.2

log(ξ*)

log(

|φ−

φ h|/|φ|

)

L2 norm; Wave direction = 20º; ξo = 31.62

α1 = α

2 = 0.5

α1 = α

2 = α

θ; θ = 0

α1 = α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

Interpolant

L2 Best Approx.

H1 Best Approx.

(a)

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2−2.5−2.4−2.3−2.2−2.1

−2−1.9−1.8−1.7−1.6−1.5−1.4−1.3−1.2−1.1

−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.2

log(ξ*)

log(

|φ−

φ h|/|φ|

)

H1 semi−norm; Wave direction = 20º; ξo = 31.62

α1 = α

2 = 0.5

α1 = α

2 = α

θ; θ = 0

α1 = α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

Interpolant

L2 Best Approx.

H1 Best Approx.

(b)

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2−2.5−2.4−2.3−2.2−2.1

−2−1.9−1.8−1.7−1.6−1.5−1.4−1.3−1.2−1.1

−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.2

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0.5

α1 = α

2 = α

θ; θ = 0

α1 = α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

Interpolant

L2 Best Approx.

H1 Best Approx.

(c)

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2−2.5−2.4−2.3−2.2−2.1

−2−1.9−1.8−1.7−1.6−1.5−1.4−1.3−1.2−1.1

−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.2

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0.5

α1 = α

2 = α

θ; θ = 0

α1 = α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

Interpolant

L2 Best Approx.

H1 Best Approx.

(d)

Figure 66: Convergence of the relative error considering ξo = 10√10 and for mesh resolutions

that guarantee at least n elements per wavelength, where n ∈ 4, 6, 8, 10, 12, 14, 16.The considered norms and the wave directions are: (a) L2 norm, β = (π/9) ; (b) H1

semi-norm, β = (π/9) ; (c) L2 norm, β = (π/4) and (d) H1 semi-norm, β = (π/4)


−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2−2.5−2.4−2.3−2.2−2.1

−2−1.9−1.8−1.7−1.6−1.5−1.4−1.3−1.2−1.1

−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.2

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0.5

α1 = α

2 = α

θ; θ = 0

α1 = α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

Interpolant

L2 Best Approx.

H1 Best Approx.

(a)

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2−2.5−2.4−2.3−2.2−2.1

−2−1.9−1.8−1.7−1.6−1.5−1.4−1.3−1.2−1.1

−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.2

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0.5

α1 = α

2 = α

θ; θ = 0

α1 = α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

Interpolant

L2 Best Approx.

H1 Best Approx.

(b)

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2−2.5−2.4−2.3−2.2−2.1

−2−1.9−1.8−1.7−1.6−1.5−1.4−1.3−1.2−1.1

−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.2

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0.5

α1 = α

2 = α

θ; θ = 0

α1 = α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

Interpolant

L2 Best Approx.

H1 Best Approx.

(c)

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2−2.5−2.4−2.3−2.2−2.1

−2−1.9−1.8−1.7−1.6−1.5−1.4−1.3−1.2−1.1

−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.2

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0.5

α1 = α

2 = α

θ; θ = 0

α1 = α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

Interpolant

L2 Best Approx.

H1 Best Approx.

(d)

Figure 67: Convergence of the relative error considering ξo = 100 and for mesh resolutionsthat guarantee at least n elements per wavelength, where n ∈ 4, 6, 8, 10, 12, 14, 16.The considered norms and the wave directions are: (a) L2 norm, β = (π/9) ; (b) H1

semi-norm, β = (π/9) ; (c) L2 norm, β = (π/4) and (d) H1 semi-norm, β = (π/4)


−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

log(ξ*)

log(

|φ−

φ h|/|φ|

)

l∞ Euclidean norm; Wave direction = 20º; ξo = 31.62

α1 = α

2 = 0.5

α1 = α

2 = α

θ; θ = 0

α1 = α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

L2 Best Approx.

H1 Best Approx.

(a)

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0.5

α1 = α

2 = α

θ; θ = 0

α1 = α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

L2 Best Approx.

H1 Best Approx.

(b)

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0.5

α1 = α

2 = α

θ; θ = 0

α1 = α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

L2 Best Approx.

H1 Best Approx.

(c)

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0.5

α1 = α

2 = α

θ; θ = 0

α1 = α

2 = α

θ; θ = π/8

α1 ≠ α

2; QSFEM

L2 Best Approx.

H1 Best Approx.

(d)

Figure 68: Convergence of the relative error in the l∞ Euclidean norm using: (a) ξo = 10√10,

β = (π/9) ; (b) ξo = 100, β = (π/9) ; (c) ξo = 10√10, β = (π/4) ; (d) ξo = 100,

β = (π/4). The considered mesh resolutions guarantee at least n elements perwavelength, where n ∈ 4, 6, 8, 10, 12, 14, 16.


the QSFEM on square meshes. Nevertheless it uses a quadratic bubble perturbationfunction defined over a macro-element and the parameters multiplying these bubblesare found by solving local optimization problems involving a functional of the localtruncation error. Alternate methods that achieve this objective were proposed earlierwithin a variational setting and with similar implementation/computational cost, viz.the RBFEM [160], the DGB method [129], the GPR method [51] etc. Can this path toobtain the QSFEM be simplified? That is the outlook of this chapter.

Recall that the nonstandard compact stencil studied here has an additional structurethat reduces its abstractness. This additional structure throws light on the extensionof this stencil to unstructured meshes. In chapter 5 and [138] a new Petrov–Galerkinmethod involving two parameters viz. α1,α2 is presented which yields this non-standard compact stencil on rectangular meshes. Making the two parameters equal,i. e.α1 = α2 = α, we recover the compact stencil obtained by the α-interpolation of theGalerkin FEM and the classical central FDM. This Petrov–Galerkin method providesthe counterparts of these two schemes on unstructured meshes and allows the treat-ment of natural boundary conditions (Neumann or Robin) and the source terms ina straight-forward manner. This we believe would open door to design higher-orderPetrov–Galerkin methods which can be an alternative to the existing higher-ordermethods for the Helmholtz equation.

In general, ideas and experience shall interact permanently.

— Lothar Collatz.

5P E T R O V– G A L E R K I N F O R M U L AT I O N

5.1 introduction

This chapter is a continuation of chapter 41 wherein a simple domain-based higher-

order compact numerical scheme involving two parameters viz. α1,α2 was presentedfor the Helmholtz equation. The stencil obtained by choosing the parameters as dis-tinct, i. e.α1 6= α2 was denoted therein as the ‘nonstandard compact stencil’. Takingα1 = α2 = α , the nonstandard compact stencil simplifies to the α-interpolationof the equation stencils obtained by the Galerkin finite element method (FEM) andclassical central finite difference method (FDM). For the Helmholtz equation, genericexpressions for the parameters were given that guarantees a dispersion accuracy ofsixth-order should α1 6= α2 and fourth-order should α1 = α2. As the findings re-ported therein and the corresponding analysis was done for compact stencils, thecontribution of the Galerkin FEM to the equation stencil corresponds to the choiceof the lowest order rectangular block finite elements. By blocks we mean Cartesianproduct of intervals and by lowest order we refer to multilinear finite-element (FE)interpolation on these blocks. In this chapter we extend this scheme to unstructuredmeshes. We shall only consider the lowest order finite elements, i. e., linear FE inter-polation on simplices and multilinear FE interpolation on blocks. The focus of thischapter is twofold: a) to design a Petrov–Galerkin (PG) method that reproduces onstructured meshes the aforesaid numerical scheme and b) to study if this PG methodusing lowest order FE on unstructured meshes inherits the higher-order dispersionaccuracy observed for the same on structured meshes.

Consider the multidimensional Helmholtz equation subjected to the following bound-ary conditions:

Lφ := −∆φ− ξ2oφ = f(x) in Ω (5.1a)

φ = φp on ΓD (5.1b)

n ·∇φ−Mφ = q on ΓR (5.1c)

where ξo is the wavenumber, f(x) is the source term, φp is the prescribed value of φ onthe Dirichlet boundary ΓD. The operator M models either the Dirichlet-to-Neumann(DtN) map should the boundary-value problem (BVP) be posed on a domain with anexterior DtN boundary or the Neumann/Robin boundary conditions should the BVPbe posed on an interior domain. Consider the following case: an interior BVP withonly Dirichlet boundary conditions and f(x) = 0; discretization of Eq.(5.1) is done byusing either the Galerkin FEM or the FDM; a structured rectangular mesh/grid in 2d

and bilinear FE interpolations on them are used. For the considered case we use the

1 Henceforth all references to chapter 4 will be cited as [137] which is the published version of the same.

129

130 5 petrov–galerkin formulation

following notation (described earlier in §3.1) to represent a generic compact stencilcorresponding to any interior node (i, j) of a structured mesh/grid.

j+1, j, j−1Ai−1, i, i+1t = 0 (5.2)

where A represents the matrix of the stencil coefficients. The equation stencil for theGalerkin FEM method corresponding to any interior node (i, j) can be written asEq.(5.2) with the following definition of the stencil coefficient matrix (A):

Afem :=`26`1

1, 4, 1t −1, 2,−1+`16`2

−1, 2,−1t 1, 4, 1

−ξ2o`1`236

1, 4, 1t 1, 4, 1(5.3)

The stencil for the classical FDM method corresponding to any interior node (i, j)can be written as Eq.(5.2) with the following definition of A:

Afdm :=`26`1

0, 6, 0t −1, 2,−1+`16`2

−1, 2,−1t 0, 6, 0

−ξ2o`1`236

0, 6, 0t 0, 6, 0(5.4)

Naturally, the first step is to ascertain the feasibility of a PG method that reproduceson structured meshes the equation stencil corresponding to the classical FDM. Notethat one may arrive at Eq.(5.4) from Eq.(5.3) by lumping the 1d mass type distribu-tions within the equation stencil wherever they occur, i. e.replacing 1, 4, 1t −1, 2,−1by 0, 6, 0t −1, 2,−1. Following this observation we may pose some questions whichwill mark the turning points of the current exposition: a) Is it possible to reproducethe mass matrix lumping technique within a PG setting ? b) Will the use of such testfunctions/weights for the current problem reproduce the FDM stencil and further theα-interpolation of the FEM and FDM stencils ? c) What is the possibility to recoverthe nonstandard compact stencil? and d) Will a direct extension of these weights tounstructured FEs result in a PG method that inherits the higher-order dispersion accu-racy observed for the aforesaid numerical scheme ? These questions will be addressedin the subsequent sections.

This chapter is organized as follows. In Section 5.2 we recover the effect of masslumping within a PG setting. The weights that incorporate this effect are also shownto lump the 1d mass type distributions within the equation stencil as discussedabove. These weights are found to be discontinuous at the inter-element boundaryand integration-by-parts needs to be done for an integral form of Eq.(5.1a) involvingthese weights. In Section 5.3 we present the variational setting needed to provide awell defined weak form for the problem. In Section 5.4.2 considering lowest orderblock FEs, we provide models for the weights on the inter-element boundaries thatwould recover the nonstandard compact stencil on structured meshes. Some examplesare presented in Section 5.6 which illustrate the pollution effect through convergencestudies in the L2 norm, the H1 semi-norm and the l∞ Euclidean norms. Finally inSection 5.7 we arrive at some conclusions.

5.2 mass lumping

Attempts to arrive at mass lumping procedures within a variational setting is not new.‘Legal’ mass lumping emanating from residual-free bubbles had been presented in


[64, 65]. In this section we try to recover the mass matrix lumping process via a PGapproach. In other words, the objective is to design the test functions Wa such thatthe following equation holds:∫

Ω

WaNb dΩ = MabL (5.5)

where, ML is the lumped mass matrix. For the sake of simplicity, the weights aredesigned to be piecewise polynomials of the same degree as that of the FE shapefunctions. This choice allows us to express the test functions in terms of the FE shapefunctions as:

Wa := WabNb (5.6)

where, W is a matrix of constant coefficients. Using Eq.(5.6), the solution to the prob-lem given by Eq.(5.5) can be expressed as follows.∫Ω

WaNb dΩ = Wac

∫Ω

NcNb dΩ = WM ; WM = ML ⇒ W = MLM−1 (5.7)

The standard row lumping technique used to obtain ML from the consistent massmatrix M can be expressed as follows.

MabL := δab

∑c

Mac = δab∑c

∫Ω

NaNc dΩ = δab∫Ω

Na dΩ (5.8)

The fact that the finite element (FE) shape functions Na being a partition of unityis used to arrive at the last part of Eq.(5.8). If ML be obtained via the row lumpingtechnique, then the test functions defined by Eq.(5.6) also form a partition of unity.This statement can be verified as follows:∑

a

Wa =∑a

WabNb =∑a

MacL M−cbNb =

∑a

MacM−cbNb

=∑a

δabNb =∑a

Na = 1(5.9a)

∑a

Wa =∑a

Na = 1 (5.9b)

The process of defining test functions Wa via Eqs.(5.6) and (5.7) is generic upto to the feasibility of a suitable lumping technique. Unfortunately, techniques otherthan the row lumping procedure generally will not render these test functions as apartition of unity. Thus, further developments in this line are restricted to those finiteelements where the row lumping procedure makes sense, i. e.linear FE interpolationon simplices and multilinear FE interpolation on blocks. Thus for these FE discretiza-tion of the solution, PG test functions can be designed that would result in the masslumping operation. Considering the fact that the test functions can model constants(as they are a partition of unity) and the exposition is done within the framework ofPG methods, consistency and conservation properties are guaranteed.

The design problem for the test functions given by Eq.(5.5) may be posed either atthe global or local level. Posing the problem at the global level results in weights that


Shape functions Na Test functions Wa Remarks

1+ ξaξ

2

1+ 3ξaξ

21d linear FE. ξa = −1, 1

(1+ ξaξ2

)(1+ ηaη2

) (1+ 3ξaξ2

)(1+ 3ηaη2

)2d rectangular bilinear FE.ξa = −1, 1, 1,−1),ηa = −1,−1, 1, 1)

(1− ξ− η), ξ,η (3− 4ξ− 4η), (4ξ− 1), (4η− 1) 2d linear triangle FE.

Table 3: Test functions corresponding to some finite elements

are piecewise continuous (C0 functions) but with a non-local support. This observa-tion is straightforward given the following facts: the matrix W is full and the shapefunctions Na being piecewise continuous. Thus the sparse structure of the resultingalgebraic system is lost. Also the space spanned by these test functions will alwaysbe non-zero on the domain boundary. This poses difficulty in providing a simpleand well-defined weak formulation for the problem subjected to Dirichlet boundaryconditions.

On the other hand, posing the problem at the local (element) level will result inweights with a local support but with a loss in C0 continuity at the element edges.Note that in this approach the global weights (assembled in a piecewise manner)are no longer a linear combination of the global FE shape functions. Henceforth therestriction of the test functions Wa to the element interiors and edges will be denotedby Wa and Wa respectively. We remark that in this approach the solution to Eq.(5.5)posed at the element level will be used only to define Wa. This is done for tworeasons, viz. a) to ensure that the test functions be a partition of unity on the elementedges and b) to be able to model Wa such that we recover the sparsity pattern of theGalerkin FEM. The later condition also allows us to construct test function spaces thatvanish at the Dirichlet boundary.

In this work we have opted for the later approach, i. e.the design problem for theweights is posed at the element level. The weights Wa corresponding to three differentelement types is listed in Table 3. Figure 69 illustrates the construction of the weightscorresponding to the 1d linear FE shape functions. Note the loss of C0 continuity atthe element edges in Fig.69b.

Consider the diffusion term∫K∇Wa ·∇Nb dΩ for an element K of a rectangular

FE mesh in 2d. Using the test functions Wa defined in Table 3 we get,∫K

∇Wa ·∇Nb dΩ =

∫K

∂Wa

∂x

∂Nb

∂xdΩ+

∫K

∂Wa

∂y

∂Nb

∂ydΩ (5.10a)

=3`22`1

1 −1 0 0

−1 1 0 0

0 0 1 −1

0 0 −1 1

+3`12`2

1 0 0 −1

0 1 −1 0

0 −1 1 0

−1 0 0 1

(5.10b)


ξ−1 0 +1

−1

−0.5

0

0.5

1

1.5

2

N2N1

W 2W 1

(a)

ξi− 1 i i+ 1

−1

−0.5

0

0.5

1

1.5

2W i

(b)

Figure 69: Test functions corresponding to the 1d linear FE that results in mass lumping. (a)Comparison of the weights Wa within an element and with respect to the 1d FEshape functions Na. (b) Illustration of the weight corresponding to an arbitrarynode i assembled element-wise. The open circles in these illustrations signify thatthe weights have not yet been defined at these points.

The stencil coefficient matrix Ad corresponding to the assembly of the above elementmatrices can be expressed as follows:

Ad =3`2`1

0, 1, 0t −1, 2,−1+3`1`2

−1, 2,−1t 0, 1, 0 (5.11)

Note that except for a scalar multiple (here ‘3’) the stencil coefficient matrix Ad isthe same as that corresponding to the diffusive term in Afdm. Ad differs here by ascalar multiple because we have not yet considered the jumps in the values of the testfunctions at the element edges. Nevertheless this example sheds light on the first twoquestions raised in Section 5.1, i. e.using these weights the 1d mass type distributionsfound in the equation stencils are lumped, thus opening way to design FEM basedPG methods that would yield the FDM stencil. Naturally, the next step is to ascertainthe feasibility of a model for Wa which when used together with Wa within a well-defined variational setting would end up in the classical FDM.

Recall that to arrive at the form∫K∇Wa ·∇Nb dΩ integration by parts needs to

be done for an integral form of Eq.(5.1a) containing discontinuous test functions. Thisis the distinction of the current work from existing stabilized FEM based PG meth-ods that follows the theoretical framework originally proposed for the Streamline–Upwind/Petrov–Galerkin (SUPG) method [92]. Thus the framework of Discontinuous–Galerkin (DG) methods is most appropriate to provide a variational setting for thecurrent work. The distinction (apart from the trivial one2) of the current work withthe DG methods is illustrated via a schematic representation of the same in Fig. 70.Fig. 70a illustrates a generic DG method. Recall that the weights on either side of anelement edge in a DG method are not only discontinuous but also independent. Thesame applies to the trial solutions (φ) and in addition to this, models φ for φ arespecified on the element edges. For conservative DG methods, φ which is sometimesnamed as the scalar numerical flux, is single valued on the element edges [4]. On theother hand, Fig. 70b illustrates the current PG method. Note that the test functions

2 DG belongs to the class of Galerkin methods where the test functions and trial solutions belong to thesame function space


Test function, w

w1

w2

K1 K2E

Trial solution, φ

φ1

φ2

K1 K2E

φ1

φ2

(a)

Test function, w

w

w

K1 K2E

w

Trial solution, φ

φφ

K1 K2E(b)

Figure 70: Comparison of the test function w and trial solution φ of a generic Discontinuous–Galerkin (DG) method with those of the current Petrov–Galerkin (PG) method.Schematic representations of w and φ for (a) a DG method and (b) the currentPG method. Note that unlike for the DG method, the w and φ for the current PGmethod are not independent on either sides of the edge E.

(w) remain discontinuous but they are no longer independent. In addition to this, wealso specify single-valued models w for w on the element edges. The trial solutionsfor the current PG method are the standard FE solutions which are C0-continuousand are not independent on either sides of the element edge. In the next section, fol-lowing the framework of DG methods and the notations used therein [4], we presentthe variational setting for the current PG method.

5.3 variational setting

In this section we present the variational setting for PG methods with discontinuoustest functions (C−1 functions) and standard finite-element trial solutions (C0 func-tions) which can be schematically represented as in Fig. 70b. First, we recall somestandard notations which are used in the developments that follows. Let Th = K bea regular family of elements K generating a partition of Ω and the summation overall elements will be indicated by

∑K. The piecewise integral

∑K

∫K will be denoted

by∫Th

. The collection of all element edges (including edges on the boundary) willbe written as Eh = E. The set of internal and boundary edges will be denoted byEoh = Eo and E∂h = E∂ respectively. The piecewise integral

∑E

∫E will be written as∫

Eh, using

∫Eoh

and∫E∂h

when the edges are interior or on the boundary respectively.The restriction of a function ϕ to K is denoted by ϕ|K.

Suppose now that the elements K1 and K2 share an element edge E, and let n1

and n2 be the normals to E exterior to K1 and K2 respectively. For an arbitrary scalarfunction ϕ, possibly discontinuous across E, the jump and the average operators aredefined as follows.

[[ϕ]] := n1ϕ|∂K1∩E + n2ϕ|∂K2∩E (5.12a)

ϕ :=1

2

(ϕ|∂K1∩E +ϕ|∂K2∩E

)(5.12b)


whereas for an arbitrary vector function σ these operators are defined as follows.

[[σ]] := n1 ·σ|∂K1∩E + n2 ·σ|∂K2∩E (5.13a)

σ :=1

2

(σ|∂K1∩E +σ|∂K2∩E

)(5.13b)

Note that ϕ and [[σ]] are scalar quantities while [[ϕ]] and σ are vectors per-pendicular to the edge E. To deal with the sums of the form

∑K

∫∂Kϕ(n ·σ), we use

the average and jump operators defined in Eq.(5.12) and Eq.(5.13). A straightforwardcomputation leads to the following formula:∑

K

∫∂K

ϕ(n ·σ) dΓ =

∫Eoh

(ϕ[[σ]] + [[ϕ]] · σ

)dΓ +

∫E∂h

ϕ(n ·σ) dΓ (5.14)

Consider an arbitrary element K with boundary ∂K and define two sub-domainswithin it viz. Ko and Kε as shown in Fig 71a. The boundary that Kε shares withKo is denoted by ∂Ko. The external normals to ∂K and ∂Ko are denoted by n andno+ respectively. The normal no− := −no+. The piecewise integral

∫Ko

+∫Kε

will bedenoted by

∫Kh

. The free parameter ε characterizes the width of the Kε sub-domain.As shown in Fig. 71b we split the definition of the test function w over the sub-domains as follows:

w|Ko := w ; w|Kε := τ (5.15)

Likewise, we split the definition of w on the respective boundaries as follows:

w|∂Ko := w|∂Ko = τ|∂Ko ; w|∂K := w (5.16)

Thus, as shown in Fig. 71b, letting ε→ 0 we arrive at a class of test functions whichwere represented schematically in Fig. 70b. Following this line, the strategy to presentthe weak form of the problem would be to first present it assuming ε to be finite andthen taking the limit ε→ 0.

Ko

Kε

n

∂K

no+

no−

∂Ko

(a)

wτ

Ko Kε∂Ko ∂K

w

εlimε→0

w

K ∂K

w

(b)

Figure 71: Schematic diagrams of an arbitrary element K ∈ Th and a test function defined overit. (a) The element K is further divided into two sub-domains Ko and Kε. (b) Thetest function w is defined piecewise as follows: w over Ko, τ over Kε, w|∂Ko = τ|∂Koand w on ∂K. The free parameter ε characterizes the width of the Kε domain.


We now proceed to present a well-defined weak formulation of the continuousproblem 5.1 wherein the test functions could possibly be discontinuous across Eh.Multiplying Eq.(5.1a) by test functions3 w and integrating formally on K, we get∫

K

w(Lφ− f) dΩ = −

∫K

w∆φ dΩ−

∫K

w(ξ2oφ+ f) dΩ (5.17a)

= −

∫Ko

w∆φ dΩ−

∫Kε

τ∆φ dΩ−

∫K

w(ξ2oφ+ f) dΩ (5.17b)

Integrating by parts the terms w∆φ and τ∆φ we get,∫K

w(Lφ− f) dΩ =

∫Ko

(∇w ·∇φ) dΩ+

∫Kε

(∇τ ·∇φ) dΩ−

∫K

w(ξ2oφ+ f) dΩ

−

∫∂Ko

w(no+ ·∇φ) dΓ −∫∂Ko

τ(no− ·∇φ) dΓ −∫∂K

w(n ·∇φ) dΓ (5.18)

Using the formula given by Eq.(5.14) on the boundary ∂Ko we get,∫K

w(Lφ− f) dΩ =

∫Kh

(∇w ·∇φ) dΩ−

∫K

w(ξ2oφ+ f)φ dΩ

−

∫∂Ko

(w[[∇φ]] + [[w]] · ∇φ

)dΓ −

∫∂K

w(n ·∇φ) dΓ (5.19)

The variational statement of the problem obtained by using the method of weightedresiduals is: Find φ ∈ U, such that ∀w ∈ V we have∫

Ω

w(Lφ− f) dΩ+

∫ΓR

w(n ·∇φ−Mφ− q) dΓ = 0 (5.20)

Using Eq.(5.19) and the following equivalent (≡) expressions:∫Ω ≡

∫Th

:=∑K

∫K

and∫E∂hw (?) dΓ ≡

∫ΓRw (?) dΓ , the variational statement can be rewritten as follows:∫

Th

w(ξ2oφ+ f) dΩ+

∫ΓR

w(Mφ+ q) dΓ =∑K

∫Kh

(∇w ·∇φ) dΩ

−∑K

∫∂Ko

(w[[∇φ]] + [[w]] · ∇φ

)dΓ −

∫Eoh

(w[[∇φ]] + [[w]] · ∇φ

)dΓ

(5.21)

We obtain the weak form of the continuous problem 5.1 by invoking the continu-ity requirement [[∇φ|Eh ]] = 0 for the solution φ in Eq.(5.21). At this point we alsoinvoke the continuity requirements for the considered test functions w in Eq.(5.21),i. e.[[w|∂Ko ]] = 0 and [[w]] = 0. The weak form thus obtained is specific to this particularchoice of test functions.∫

Th

w(ξ2oφ+ f) dΩ+

∫ΓR

w(Mφ+ q) dΓ =∑K

∫Kh

(∇w ·∇φ) dΩ (5.22)

After any appropriate discretization, the weak form of the discretized problem isobtained by replacing the continuous unknowns by their discrete counterparts,∫

Th

wh(ξ2oφh + f) dΩ+

∫ΓR

wh(Mφh + q) dΓ =∑K

∫Kh

(∇wh ·∇φh) dΩ (5.23)

3 Whenever the solution φ is complex-valued, the complex-conjugate of w is used instead.


In the above discrete weak form the continuity constraint [[∇φh|Eoh ]] = 0 is enforcedin a weak way. Consider the term ∇wh ·∇φh in Eq.(5.23) and re-integrating by partsboth the terms in the piecewise integral

∫Kh

we get,

∫Kh

(∇wh ·∇φh) dΩ =

∫Ko

(∇wh ·∇φh) dΩ+

∫Kε

(∇τh ·∇φh) dΩ (5.24a)

= −

∫Ko

wh∆φh dΩ−

∫Kε

τh∆φh dΩ+

∫∂K

wh(n ·∇φh) dΓ (5.24b)

= −

∫Kh

wh∆φh dΩ+

∫∂K

wh(n ·∇φh) dΓ (5.24c)

To arrive at Eq.(5.24b) we have used the following results for the boundary ∂Ko:[[wh|∂Ko ]] = 0 and [[∇φh|∂Ko ]] = 0. Using the result in Eq.(5.16) and re-integrating byparts only the ∇τh ·∇φh term in the piecewise integral

∫Kh

of Eq.(5.23) we get,∫Kh


∫Ko

(∇wh ·∇φh) dΩ−

∫Kε

τh∆φh dΩ

−

∫∂Ko

wh(no+ ·∇φh) dΓ +∫∂K

wh(n ·∇φh) dΓ(5.25)

As the parameter ε→ 0 we see that no+ → n, no− → −n and ∂Ko → ∂K. Also notethat ∀ϕ ∈ L2(K),

∫Koϕ dΩ →

∫Kϕ dΩ and

∫Kεϕ dΩ → 0. Thus, taking the limit

ε→ 0, we have:

limε→0

∫Kh


∫K

(∇wh ·∇φh) dΩ+ limε→0

∫Kε

(∇τh ·∇φh) dΩ

(5.26a)

= −

∫K

wh∆φh dΩ+

∫∂K

wh(n ·∇φh) dΓ (5.26b)

=

∫K

(∇wh ·∇φh) dΩ+

∫∂K

(wh − wh)(n ·∇φh) dΓ (5.26c)


Thus in the limit ε→ 0, the discrete weak form given by Eq.(5.23) can be expressedequivalently by the following forms:∫

Th

wh(ξ2oφh + f) dΩ+

∫ΓR

wh(Mφh + q) dΓ =∑K

limε→0

∫Kh

(∇wh ·∇φh) dΩ

(5.27a)

=

∫Th

(∇wh ·∇φh) dΩ+∑K

limε→0

∫Kε

(∇τh ·∇φh) dΩ (5.27b)

= −

∫Th

wh∆φh dΩ+∑K

∫∂K

wh(n ·∇φh) dΓ (5.27c)

= −

∫Th

wh∆φh dΩ+

∫Eoh

wh[[∇φh]] dΓ +∫ΓR

wh(n ·∇φh) dΓ (5.27d)

=

∫Th

(∇wh ·∇φh) dΩ+∑K

∫∂K

(wh − wh)(n ·∇φh) dΓ (5.27e)

=

∫Th

(∇wh ·∇φh) dΩ−

∫Eoh

(wh − wh[[∇φh]] + [[wh]] · ∇φh

)dΓ

−

∫E∂h

wh(n ·∇φh) dΓ +∫ΓR

wh(n ·∇φh) dΓ(5.27f)

Note that in the limit ε → 0 the test functions wh develop sharp Dirac-type layersat the element boundaries Eh. Hence the integral

∫Kε

that appears in Eq.(5.27a) andEq.(5.27b) does not vanish as ε → 0. On the other hand, from Eq.(5.27c) we note thatthe sparsity of the resulting discrete system depends on the employed model for wh.Clearly, if the value of wh be designed to be zero on the boundary of a patch ofelements associated to any given node, then from Eq.(5.27c) we see that the resultingdiscrete system will have a sparsity pattern equivalent to that of the Galerkin FEM.Also note that on a generic block finite element, the wh∆φh term will not vanish andneeds to be evaluated. Nevertheless, using the weak form expressed as in Eq.(5.27e)the extra labor just involves the evaluation of the element boundary integrals. Thiscan be easily incorporated within an ‘assemble-by-elements’ data structure.

5.4 block finite elements

5.4.1 1d linear FE

In this section we will provide models for the PG weights on the elements edges.Consider the 1d linear FE and corresponding PG weights specification illustrated inFig. 69. In Fig. 69b, let Wi|i−1, Wi|i and Wi|i+1 be the models for the weight Wi onthe edges i− 1, i and i+ 1 respectively. For these weights Wi to be a partition of unityalso on the element edges the following relation should hold:

Wi|i−1 + Wi|i + W

i|i+1 = 1 (5.28)

There exists an infinity of solutions for Eq.(5.28) but only the choice Wi|i−1, Wi|i,Wi|i+1 = 0, 1, 0 will result in a discrete system that has the same sparsity structureas that of the Galerkin FEM or the classical FDM. Also, the space spanned by these


weights can be restricted to zero on the Dirichlet boundary without being triviallyzero inside the domain and thus, justifying their admittance in weak formulations.Using the later definition the PG weights corresponding to the 1d linear FEs can berepresented as shown in Fig. 72

ξ−1 0 +1

−1

−0.5

0

0.5

1

1.5

2W 1 W 2

(a)

ξi− 1 i i+ 1

−1

−0.5

0

0.5

1

1.5

2W i

(b)

Figure 72: A model for the PG weights on the element edges corresponding to the 1d linearFE. (a) Illustration of the weights Wa within an element. (b) The weight corre-sponding to an arbitrary node i assembled element-wise. The filled circles in theseillustrations represent the chosen model for the weights on the element edges.

Discretization of the space by finite elements will lead to the approximation φh =

NaΦa and using the PG weights (shown in Fig. 72) in any of the weak forms pre-sented in Eq.(5.27), the following equation stencil is obtained:

(1`

)(−Φi−1 + 2Φi −Φi+1) − ξ2o`Φ

i = 0 (5.29)

This is precisely the equation stencil obtained by using either the classical FDM orthe Galerkin FEM wherein the mass matrix is lumped. Define a free parameter α andconsider the following definition of the PG weights within an element:

Waα =

α(1+ 3ξξa

2

)+ (1−α)

(1+ ξξa2

)=(1+ (1+ 2α)ξξa

2

)−1 < ξ < 1

(1+ ξξa2

)ξ = ±1

(5.30)

In the above equation choosing α = 0 we will recover the weights Wa0 as the stan-

dard 1d FE shape functions Na. Likewise, choosing α = 1, the obtained weights Wa1

is the same as shown in Fig. 72. Using the PG weights defined by Eq.(5.30) in any ofthe weak forms presented in Eq.(5.27), the following equation stencil is obtained:

(1−α)

[(1

`

)(−Φi−1 + 2Φi −Φi+1) −

(ξ2o`

6

)(Φi−1 + 4Φi +Φi+1)

]

+α

[(1

`

)(−Φi−1 + 2Φi −Φi+1) − ξ2o`Φ

i

]= 0

(5.31)

⇒(1

`

)(−Φi−1 + 2Φi −Φi+1) − (1−α)

(ξ2o`

6

)(Φi−1 + 4Φi +Φi+1)

−αξ2o`Φi = 0

(5.32)


Equation (5.31) is precisely the α-interpolation of the stencils obtained by the GalerkinFEM and the classical FDM methods in 1d. For the 1d case using linear FE and asshown in Eq.(5.32), it is equivalent to the Galerkin FEM wherein an alpha-interpolatedmass matrix is used.

5.4.2 2d bilinear FE

Consider now the 2d bilinear rectangular FE and define the PG weights over theseblocks as the Cartesian product of its 1d counterparts defined earlier in Eq.(5.30).Thus we get,

Waα =

(1+ (1+ 2α)ξξa

2

)(1+ (1+ 2α)ηηa

2

)(ξ,η) ∈ (−1, 1)× (−1, 1)

(1+ (1+ 2α)ξξa

2

)(1+ ηηa2

)(ξ,η) ∈ (−1, 1)× ±1

(1+ ξξa2

)(1+ (1+ 2α)ηηa

2

)(ξ,η) ∈ ±1× (−1, 1)

(1+ ξξa2

)(1+ ηηa2

)(ξ,η) ∈ ±1× ±1

(5.33)

Note that on the element edges whenever Na = 0, we have simultaneously Wa =

0. Likewise, on the edges whenever the expression for α is single-valued, we havesimultaneously a single-valued model for Wa. In this way it is possible to retain thesparsity pattern of the Galerkin FEM. On a structured mesh in 2d and using the PGweights defined by Eq.(5.33) in any of the weak forms presented in Eq.(5.27), thestencil corresponding to any interior node (i, j) can be written as Eq.(5.2) with thefollowing definition of the stencil coefficient matrix (A):

Aα := (1−α)`26`1

1, 4, 1t −1, 2,−1+α`26`1

0, 6, 0t −1, 2,−1

+ (1−α)`16`2

−1, 2,−1t 1, 4, 1+α`16`2

−1, 2,−1t 0, 6, 0

− (1−α)ξ2o`1`236

1, 4, 1t 1, 4, 1−αξ2o`1`236

0, 6, 0t 0, 6, 0

(5.34)

This is precisely the α-interpolation of the FEM and FDM stencils in 2d, i. e.Aα =

(1− α)Afem + αAfdm. Unlike in 1d where we had a unique way to model Wa so asto retain the sparsity pattern of the Galerkin-FEM, in 2d many alternatives modelsexits. However, all acceptable models for Wa have to be a partition of unity for everyelement and be single-valued on the element edges. In the following we show oneof the many possible models for Wa and for an unstructured 2d bilinear block FE.Let α1,α2 be two free parameters and consider the following definition for the PGweights:

Waα1,α2 =

Waα2

:= Wabα2Nb (ξ,η) ∈ (−1, 1)× (−1, 1)

(1+ (1+ 2α1)ξξa

2

)(1+ ηηa2

)(ξ,η) ∈ (−1, 1)× ±1

(1+ ξξa2

)(1+ (1+ 2α1)ηηa

2

)(ξ,η) ∈ ±1× (−1, 1)

(1+ ξξa2

)(1+ ηηa2

)(ξ,η) ∈ ±1× ±1

(5.35)


where Wabα2

:= (1− α2)δab + α2Wab, δab is the Kronecker delta and Wab is the

matrix of constant coefficients given by Eq.(5.7). Note that Wa is designed to be apartition of unity. Choosing a single valued expression for α1 on each element edgeimplies a single-valued model for Wa on the same. Thus, should any length scaleappear within the expression for α1, then it should be proportional to the correspond-ing edge length. On a structured mesh in 2d and using the PG weights defined byEq.(5.35) in any of the weak forms presented in Eq.(5.27), the stencil corresponding toany interior node (i, j) can be written as Eq.(5.2) with the following definition of thestencil coefficient matrix (A):

Aα1,α2 := (1−α1)`26`1

1, 4, 1t −1, 2,−1+α1`26`1

0, 6, 0t −1, 2,−1

+ (1−α1)`16`2

−1, 2,−1t 1, 4, 1+α1`16`2

−1, 2,−1t 0, 6, 0

− (1−α2)ξ2o`1`236

1, 4, 1t 1, 4, 1−α2ξ2o`1`236

0, 6, 0t 0, 6, 0

(5.36)

This is precisely the nonstandard compact stencil presented in [137]. Note that tak-ing α1 = α2 = αwe recover the stencil given by Eq.(5.34), i. e.the α-interpolation of theFEM and FDM stencils in 2d: Aα = (1− α)Afem + αAfdm. Choosing α1 = α2 = 0.5we get a stencil that is the average of the FEM and FDM stencils in 2d and can beshown to be equal [137] to the stencil obtained by the generalized fourth-order com-pact Padé approximation [81, 168] (therein using the parameter γ = 2). Likewisetaking α1 = 0 and α2 = α we arrive at a stencil that results from the Galerkin FEMmethod using an α-interpolated mass matrix Mα := (1−α)M +αML.

5.4.3 Stabilization Parameters

Considering square meshes/grids (i. e.`1 = `2) the parameters α1 and α2 that ap-pear in Aα1,α2 can be chosen such that the numerical solution be sixth-order accurate,i. e.O

((ξo`)

6)

or equivalently O(ω3). Recall that this is the maximum order of accu-

racy that can be attained on any compact stencil [10]. All such α1 and α2 should obeythe following series expansion in terms of ω [137].

α1 =1

2−ω

60+

∞∑m=2

amωm ; α2 =

1

2−ω

40+

∞∑m=2

bmωm (5.37)

where am,bm are coefficients independent of ω. The relative phase P and localtruncation T errors of these schemes can be expressed as follows:

P = r3ω3 +O

(ω4)

; T = −2r3ω3 +O

(ω4)

(5.38)

r3 =

[5

110592−

(a2 − 4b248

)+

(1+ 576a227648

)cos(4β) +

cos(8β)774144

](5.39)

As am,bm (m > 2) can be chosen arbitrarily, infinitely many sixth-order schemescan be designed through Aα1,α2 . Of course some particular choice of am,bm mayyield a scheme with better features. For instance, am,bm may be chosen such thatthe local truncation error T be zero along some chosen directions. Choosing α1 =

α2 = α, i. e.am = bm ∀m we recover the α-interpolation of the Galerkin FEM and


FDM. Further details on the choice of the parameters to recover various stencils canbe found in [137].

As most of the expressions for α1,α2 optimized for square meshes need not be op-timal for unstructured meshes, in the current work we consider only the simplest ex-pressions that would guarantee fourth-order (Eq.(5.40a)) and sixth-order (Eq.(5.40b))dispersion accuracy on square meshes. On unstructured meshes the expressions forα1,α2 corresponding to these two choices can be written as follows:

α1 = α2 =1

2(5.40a)

α1 =1

2−ω

60; α2 =

1

2−ω

40(5.40b)

where ω := (ξo)2 and ω := (ξo˜)2. and ˜ represent the models used for the lengthmeasures corresponding to the element edges and the interior. In the current study foreach element we have chosen equal to the edge length (will vary from edge to edge)and ˜ equal to the maximum edge length. Note that using this model, α1 is alwayssingle-valued on the edges. On square meshes using Eq.(5.40b) we recover α1,α2 asgiven in Eq.(5.37) up to the first two terms which is sufficient to attain sixth-orderdispersion accuracy.

5.5 simplicial finite elements

Consider a rectangular domain discretized by structured simplicial FEs. Such dis-cretization would typically yield stencils as shown in figure 73. The stencils with thehypotenuse oriented along left and right are labeled using the markers o = l ando = r respectively.

ℓ1

ℓ2

ℓ1

ℓ2

(a)

ℓ1

ℓ2

ℓ1

ℓ2

(b)

Figure 73: Stencils obtained by using a structured simplicial finite element mesh with thehypotenuse oriented/tilted along (a) left, i. e.o = l and (b) right , i. e.o = r. ‘o’ is aflag that indicates the stencil tilt.

The equation stencil for the Galerkin FEM corresponding to any interior node (i, j)can be written as Eq.(5.2) with the following definition of stencil coefficient matrix:

Afem =`2`1

0 0 0

−1 2 −1

0 0 0

+

`1`2

0 −1 0

0 2 0

0 −1 0

−

ξ2o`1`212

δol 1 δor

1 6 1

δor 1 δol

(5.41)

5.5 Simplicial finite elements 143

where δol and δor are Kronecker deltas and ‘o’ is a flag that indicates the stenciltilt. Note that using simplicial FEs the contribution of the diffusion term in Eq.(5.41)is identical to that obtained in the FDM stencil given by Eq.(5.4). Thus, the stencilobtained via an α-interpolation of the Galerkin FEM and the FDM stencils will leadto the following stencil coefficient matrix:

Aα =`2`1

0 0 0

−1 2 −1

0 0 0

+

`1`2

0 −1 0

0 2 0

0 −1 0

−

ξ2o`1`212

(1−α)δol (1−α) (1−α)δor

(1−α) 6(1+α) (1−α)

(1−α)δor (1−α) (1−α)δol

(5.42)

We see that using simplicial FEs in 2d, the α-interpolation of Galerkin FEM andFDM is equivalent to the alpha-interpolation method (AIM) [110, 140]. In the AIM,the consistent mass matrix M that appears in the Galerkin FEM is replaced by the α-interpolated mass matrix Mα := (1− α)M + αML. Consider the following definitionof the PG weights for simplicial FEs,

Waα =

Waα := Wab

α Nb in the element interior

Wa := Na on the element edges(5.43)

Using the above weights for simplicial FEs in any of the weak forms presentedin Eq.(5.27), we recover the AIM within a PG setting. In particular for structuredsimplicial FE meshes we recover the stencil given in Eq.(5.42). We can guess that asolution to any generic stencil takes the form Φi,j := φ(xi1, xj2) = exp[i(ξh1x

i1 + ξ

h2xj2)].

Substituting this solution into the stencil formed by Aα given in Eq.(5.42) and definingλ1 := exp(iξh1 `1) and λ2 := exp(iξh2 `2) we get the characteristic equation as follows:

[2− λ1 − λ−11 ]

ω1+

[2− λ2 − λ−12 ]

ω2=

(1+α)

2+

(1−α)

12

(λ1 + λ

−11 + λ2 + λ

−12

)

+(1−α)

12

(δol[λ1λ

−12 + λ−11 λ2] + δor[λ1λ2 + λ

−11 λ−12 ]

)(5.44)

where, ω1 := (ξo`1)2 and ω2 := (ξo`2)

2. For the dispersion analysis of Aα given inEq.(5.42) we restrict to the case `1 = `2 = `. In this case, the stencil coefficient matrixAα simplifies to,

Aα =

δolA2 A1 δorA2

A1 A0 A1

δorA2 A1 δolA2

;

A0 := 4− (1+α)(ω/2)

A1 := −1− (1−α)(ω/12)

A2 := −(1−α)(ω/12)

(5.45)

where, ω := (ξo`)2. The characteristic equation given in Eq.(5.44) now gets simpli-

fied to the following:

A0 + 2A1[cos(ξh1 `) + cos(ξh2 `)] + 2A2 cos(ξh1 `± ξh2 `) = 0 (5.46)

The ‘±’ that appears in the above equation corresponds to the cases o = r and o = l

respectively (see figure 73). The parameter α may be expressed as a generic seriesexpansion in terms of ω as follows:

α :=

∞∑m=0

amωm ≈ a0 + a1ω+ a2ω

2 + a3ω3 +O

(ω4)

(5.47)


where am,bm are coefficients independent of ω. Following the approach used in[137] which was originally presented in [10], the relative phase (P) and local trunca-tion (T) errors along any direction β can be written as,

P = r1ω+O(ω2)

; T = −2r1ω+O(ω2)

(5.48)

r1 :=(a0 − 1)

24[2± sin(2β)] +

[3+ cos(4β)

96

](5.49)

Clearly it is impossible to obtain the condition r1 = 0 by a choice of the coeffi-cient a0 that is independent of the angle β. Thus, unlike for structured bilinear blockFEs, unfortunately for structured simplicial FEs the pollution is essentially of thesame order as for those of the Galerkin FEM, the FDM and the GLS-FEM [79, 181].Nevertheless, just like for the GLS-FEM, the coefficient a0 can be chosen so as toarrive at a higher-order modification of the interior stencil of the Galerkin FEM. Sim-ilar studies for eigenvalue problems using the AIM with simplicial FEs was done in[109, 110, 139, 140].

Remark: Following the approach taken for bilinear block FEs, it is possible to providedifferent models for the PG weights on the elements edges. This idea will be exploredin future works.

5.6 examples

In this section we present some examples in 2d for the problem defined by Eq.(5.1) andconsidering the following problem data: the wavenumber ξo ∈ 50, 100, the sourcef = 0, the direction of wave propagation β = (π/9) and the domain Ω = [0, 1]× [0, 1].The domain Ω is discretized by considering both structured and unstructured meshesmade up of just the bilinear block-FEs. The unstructured meshes are obtained byrandomly perturbing the interior nodes of structured meshes with coordinates (xi,yi)as follows [60, 128]:

x′i = xi + `1δrand() ; y

′i = yi + `2δrand() (5.50)

where, (x′i,y

′i) represent the corresponding coordinates of the unstructured mesh, δ

is a mesh distortion parameter and rand() is a function that returns random numbersuniformly distributed in the interval [−1, 1]. Figure 74 illustrates an instance of anunstructured mesh obtained by this procedure using a 50× 50 square mesh and theparameter δ = 0.2.

We consider the following four cases concerned with the choice of the stabilizationparameters α1,α2:

I: α1 = α2 = 0. This case corresponds to the Galerkin FEM.

II: α1 = α2 = 1. This case on rectangular meshes corresponds to the classical FDM.We denote this case as FDM/PG as it is obtained within a Petrov–Galerkinframework. FDM/PG is a straight-forward extension of the FDM to unstruc-tured meshes.

III: α1 = α2 = (1/2). This case corresponds to a discrete system that is equivalentto the average of the Galerkin FEM and the FDM/PG. On rectangular meshes

5.6 Examples 145

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

Figure 74: Meshes made of bilinear block-FEs. (a) Structured mesh, δ = 0. (b) Unstructuredmesh, δ = 0.2.

we obtain the stencil associated with (Afem + Afdm)/2, which is equivalent tothe one obtained using the generalized Padé approximation in 2d [81, 168]. Thedispersion accuracy on square meshes is of fourth-order.

IV: α1 6= α2 6= 0 and given by Eq.(5.40b). On rectangular meshes this case yields thenonstandard compact stencil presented in [137]. Recall that on square meshesthese expressions for the parameters α1,α2, guarantee sixth-order dispersionaccuracy.

For these considerations we study the convergence of the relative error in the fol-lowing norms:

L2 norm‖φ−φh‖0‖φ‖0

:=[∫Ω |φ−φh|

2 dΩ]1/2

[∫Ω |φ|2 dΩ]1/2

(5.51a)

H1 semi-norm‖φ−φh‖1‖φ‖1

:=[∫Ω |∇(φ−φh)|

2 dΩ]1/2

[∫Ω |∇φ|2 dΩ]1/2

(5.51b)

l∞ Euclidean norm|Φe −Φh|∞

|Φe|∞ :=maxi |Φie −Φih|

maxi |Φie|(5.51c)

whereΦe is the exact solution sampled at the nodes of the mesh. In the convergencestudies done here, the numerical solutions corresponding to the four cases viz. I-IV,are compared with the following solutions: the nodally exact interpolant denoted byIhφ and the best approximations with respect to the L2 norm and the H1 semi-norm


denoted by P0hφ and P1hφ respectively. The solutions Ihφ, P0hφ and P1hφ can be foundas shown in Eq.(5.52).

Ihφ := NaΦae (5.52a)∫Ω

wh(φ− P0hφ) dΩ = 0 ∀ wh ∈ Vh0⇒ ‖ φ− P0hφ ‖0 6 ‖ φ−φh ‖0 ∀ φh ∈ Vh

(5.52b)

∫Ω

∇wh ·∇(φ− P1hφ) dΩ = 0 ∀ wh ∈ Vh0⇒ ‖ φ− P1hφ ‖1 6 ‖ φ−φh ‖1 ∀ φh ∈ Vh

(5.52c)

As the exact solution φ is sinusoidal, we have used a third-order Gauss quadraturerule to evaluate the expressions involving φ in Eq.(5.51) and Eq.(5.52).

5.6.1 Example 1: Dirichlet boundary conditions

In this example, only the Dirichlet boundary conditions are prescribed such that theexact solution is φ(x) = sin(ξβ · x), where ξβ := ξo(cos(β), sin(β)). Structured mesheswith n×n square elements are considered with n given by the following expression.

n = ceil(50× 2m/8) ; m ∈ 0, 1, 2, . . . 28 (5.53)

where ceil(m) is a function that returns the nearest integer greater than or equal tom. Unstructured meshes are obtained corresponding to each structured mesh usingthe procedure described earlier. For these considerations we present the plots of therelative error vs. the mesh-size.

Figure 75 illustrates the convergence of the relative error in the L2 norm. Clearlythe error lines of the considered solutions are bounded from below by the error lineof P0hφ (L2-BA) and show a tendency to become parallel to the error line of P0hφ as`→ 0. Figures 75a and 75b show the L2 error considering ξo = 50 and for structured(δ = 0) and unstructured (δ = 0.2) meshes respectively. As expected the error linescorresponding to cases I and II differs substantially from those of Ihφ, P0hφ and P1hφ.The error lines corresponding to cases III and IV are very close to that of Ihφ. Asthe solution in case IV has sixth-order dispersion accuracy on square meshes it isalmost the same as Ihφ. On unstructured meshes the quality of the solution in case IVdeteriorates and is similar to that of case III. Figures 75c and 75d show the error linesconsidering ξo = 100 and for the choices δ = 0 and δ = 0.2 respectively. As expectedall the error lines corresponding to cases I-IV deviate further from the error linesof Ihφ, P0hφ and P1hφ (the pollution effect). On square meshes the solution of case IVshows the least deviation and is practically identical to Ihφ (Figure 75c). The pollutionassociated with the solution of case III is similar to that of cases I and II on coarsemeshes but it diminishes rapidly on further mesh refinement. Again, on unstructuredmeshes the quality of the solution in case IV deteriorates showing an appreciabledeviation from the error lines of Ihφ, P0hφ and P1hφ and is similar to that of case III(Figure 75d). A distinctive feature in these plots is the formation of spikes in the errorlines. Their presence is more evident for higher wavenumbers and on unstructuredmeshes where the dispersion errors are relatively higher. As here we have prescribedonly the Dirichlet boundary conditions the numerical solutions might suffer spurious

5.6 Examples 147

amplitude and/or phase modulations to satisfy them [79]. Encounters with zones ofdegeneracy (wherein the solution might blow up) also contributes to huge errors inthe amplitude [47, 79, 137]. Fortunately, these spurious modulations reduce shouldother choices for the boundary conditions be employed viz. an exterior problem withDtN boundary conditions [79], an interior problem with Robin boundary conditions[10].

Figure 76 illustrates the convergence of the relative error in the H1 semi-norm.Clearly the error lines of the considered solutions are bounded from below by theerror line of P1hφ (H1-BA). Unlike the errors measured in the L2 norm, the errorsmeasured in the H1 semi-norm show a tendency to merge with the error line ofP1hφ. Figures 76a and 76b (ξo = 50) show that the error lines of case III and IV arepractically the same as of Ihφ, P0hφ and P1hφ. Figures 76c and 76d (ξo = 100) showthat the deviations of the error lines of cases III and IV from the error line of P1hφeven though they exist, it is smaller than that observed using the L2 norm.

Figure 77 illustrates the convergence of the relative error in the l∞ Euclidean normwhich is a measure of nodal exactness. Figures 77a and 77c show that on structuredmeshes (δ = 0) the error lines of case III and IV converge at a rate of fourth and sixthorder respectively. Figures 77b and 77d show that on unstructured meshes (δ = 0.2)the higher order accuracy of case IV deteriorates and has a trend similar to that of caseIII. Also, in an average sense both the cases III and IV have second-order convergencerate similar to P0hφ and P1hφ. For the wavenumber ξo = 50 the errors found for thecases III and IV are similar to that of P0hφ (Figure 77b).

5.6.2 Example 2: Robin boundary conditions

In this example, only the Robin boundary conditions are prescribed such that theexact solution is φ(x) = exp(iξβ · x), where ξβ := ξo(cos(β), sin(β)). The operator M

that appears in Eq.(5.1c) is chosen as M := iξo. Thus, q(x) := i(n ·ξβ− ξo) exp(iξβ · x).Structured meshes with n× n square elements are considered with n given by thefollowing expression.

n = ceil(mξo

2π) ; m ∈ 10, 10.5, 11, 11.5, . . . 25 (5.54)

Choosing n by the above expression guarantees the presence of at least m elementsper wavelength. Unstructured meshes are obtained corresponding to each structuredmesh using the procedure described earlier. For these considerations, we present theplots of the relative error vs. ξ∗, where ξ∗ := (ξo`/π). The choice of ξ∗ as the abscissain the plots allows us to single out the pollution effect.

Figures 78, 79 and 80 illustrate the convergence of the relative error in the L2 norm,the H1 semi-norm and the l∞ Euclidean norm respectively. Clearly, all the spuriousmodulations that appeared in the error lines considering only the Dirichlet boundaryconditions (Figures 75, 76 and 77) diminish when the Robin boundary conditionsare prescribed. Also, in all the figures (78, 79 and 80) by freezing the value of δ andincreasing the value of ξo we observe the following trait. The location of the error linesof Ihφ, P0hφ and P1hφ is practically unaffected by an increase in ξo (no pollution). Asexpected the error lines of cases I and II not only are located high above the error linesof Ihφ, P0hφ and P1hφ but also shift higher with an increase in ξo (pollution effect).


−2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

log(h)

log(

|φ−

φ h|/|φ|

)

Error in L2 norm; ξo = 50.00; δ = 0.0

α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(a)

−2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

log(h)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(b)

−2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

log(h)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(c)

−2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

log(h)

log(

|φ−

φ h|/|φ|

)Error in L2 norm; ξ

o = 100.00; δ = 0.2

α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(d)

Figure 75: Convergence of the relative error in the L2 norm using β = (π/9) and Dirichletboundary conditions. The wavenumber ξo and the mesh distortion parameter usedare: (a) ξo = 50, δ = 0 ; (b) ξo = 50, δ = 0.2 ; (c) ξo = 100, δ = 0 and (d) ξo = 100,δ = 0.2.

On uniform meshes (δ = 0) the error lines of cases III and IV not only are locatedclose to the respective best approximations but also show negligible upward shiftwith an increase in ξo (small pollution). Clearly, on uniform meshes the performanceof case IV is relatively better than that of case III (although the difference is small).The pollution effect is more visible for these cases on unstructured meshes (δ = 0.2).In the L2 norm the error lines of cases III and IV show an accuracy at par with Ihφand P1hφ (figures 78c and 78d). In the H1 semi-norm the error lines of cases III and IVare practically the same as those corresponding to Ihφ, P0hφ and P1hφ (figures 79c and

5.6 Examples 149

−2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

log(h)

log(

|φ−

φ h|/|φ|

)Error in H1 semi−norm; ξ

o = 50.00; δ = 0.0

α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(a)

−2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

log(h)lo

g(|φ

−φ h|/|

φ|)

Error in H1 semi−norm; ξo = 50.00; δ = 0.2

α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(b)

−2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

log(h)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(c)

−2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

log(h)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(d)

Figure 76: Convergence of the relative error in the H1 semi-norm using β = (π/9) and Dirich-let boundary conditions. The wavenumber ξo and the mesh distortion parameterused are: (a) ξo = 50, δ = 0 ; (b) ξo = 50, δ = 0.2 ; (c) ξo = 100, δ = 0 and (d)ξo = 100, δ = 0.2.

79d). In the l∞ Euclidean norm the error lines of cases III and IV are close to the errorline of P0hφ (figures 80c and 80d). Further, in Figure 80 note that in an average senseall the error lines have second-order convergence rate in the l∞ Euclidean norm. Thisresult is due to the error in the approximation of the Robin boundary condition. Thus,unlike in Figure 77 wherein the error lines of cases III and IV showed fourth-orderand sixth-order convergence rates respectively, here it drops to second-order.


−2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

log(h)

log(

|φ−

φ h|/|φ|

)

Error in L∞ vector norm; ξo = 50.00; δ = 0.0

α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

L2−BA

H1−BA

(a)

−2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

log(h)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

L2−BA

H1−BA

(b)

−2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

log(h)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

L2−BA

H1−BA

(c)

−2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

log(h)

log(

|φ−

φ h|/|φ|

)Error in L∞ vector norm; ξ

o = 100.00; δ = 0.2

α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

L2−BA

H1−BA

(d)

Figure 77: Convergence of the relative error in the l∞ Euclidean norm using β = (π/9) andDirichlet boundary conditions. The wavenumber ξo and the mesh distortion pa-rameter used are: (a) ξo = 50, δ = 0 ; (b) ξo = 50, δ = 0.2 ; (c) ξo = 100, δ = 0 and(d) ξo = 100, δ = 0.2.

5.7 conclusions

A new Petrov–Galerkin (PG) method involving two parameters viz. α1,α2 is pre-sented which yields the following schemes on rectangular meshes: a) a compact sten-cil obtained by the α-interpolation of the Galerkin FEM and the classical central FDM,should the two parameters be made equal, i. e.α1 = α2 = α and b) the nonstandardcompact stencil presented in [137] for the Helmholtz equation if the parameters are

5.7 Conclusions 151

−1.1 −1.05 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7−3

−2.75

−2.5

−2.25

−2

−1.75

−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

log(ξ*)

log(

|φ−

φ h|/|φ|

)Error in L2 norm; ξ

o = 50.00; δ = 0.0

α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(a)

−1.1 −1.05 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7−3

−2.75

−2.5

−2.25

−2

−1.75

−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

log(ξ*)lo

g(|φ

−φ h|/|

φ|)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(b)

−1.1 −1.05 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7−3

−2.75

−2.5

−2.25

−2

−1.75

−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(c)

−1.1 −1.05 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7−3

−2.75

−2.5

−2.25

−2

−1.75

−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(d)

Figure 78: Convergence of the relative error in the L2 norm using β = (π/9) and Robin bound-ary conditions. The wavenumber ξo and the mesh distortion parameter used are:(a) ξo = 50, δ = 0 ; (b) ξo = 100, δ = 0 ; (c) ξo = 50, δ = 0.2 and (d) ξo = 100,δ = 0.2.

distinct, i. e.α1 6= α2. On square meshes, these two schemes were shown to providesolutions to the Helmholtz equation that have a dispersion accuracy of fourth andsixth order respectively [137]. Thus, this Petrov–Galerkin method yields in a straight-forward manner the counterparts of these two schemes on unstructured meshes.

The salient features of this new PG method include the following. The solutionspace is discretized by standard C0-continuous finite elements. The test function-s/weights are piecewise polynomials of the same degree as the FE shape functionsand are generally discontinuous at the inter-element boundaries. Models for the weights


−1.1 −1.05 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7−1.25

−1.125

−1

−0.875

−0.75

−0.625

−0.5

−0.375

−0.25

−0.125

0

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(a)

−1.1 −1.05 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7−1.25

−1.125

−1

−0.875

−0.75

−0.625

−0.5

−0.375

−0.25

−0.125

0

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(b)

−1.1 −1.05 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7−1.25

−1.125

−1

−0.875

−0.75

−0.625

−0.5

−0.375

−0.25

−0.125

0

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(c)

−1.1 −1.05 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7−1.25

−1.125

−1

−0.875

−0.75

−0.625

−0.5

−0.375

−0.25

−0.125

0

log(ξ*)

log(

|φ−

φ h|/|φ|

)Error in H1 semi−norm; ξ

o = 100.00; δ = 0.2

α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

Interpolant

L2−BA

H1−BA

(d)

Figure 79: Convergence of the relative error in the H1 semi-norm using β = (π/9) and Robinboundary conditions. The wavenumber ξo and the mesh distortion parameter usedare: (a) ξo = 50, δ = 0 ; (b) ξo = 100, δ = 0 ; (c) ξo = 50, δ = 0.2 and (d) ξo = 100,δ = 0.2.

on the inter-element boundaries are provided such that the sparsity pattern is thesame as that for the Galerkin FEM. The parameters α1,α2 determine the shape of theweights on the inter-element boundaries and the element interior respectively. Thechoice α1 = α2 = 0 yield weights that are identical to the FE shape functions andhence we recover the Galekin FEM. The weights are a partition of unity only in thesense that they add up to unity. As the row lumping technique for the FEM massmatrices is a critical step in the design of these weights (to fulfill the partition of unityconstraint), the current PG method is restricted only to those FEs where this tech-

5.7 Conclusions 153

−1.1 −1.05 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7−3

−2.75

−2.5

−2.25

−2

−1.75

−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

0.25

0.5

log(ξ*)

log(

|φ−

φ h|/|φ|

)Error in L∞ vector norm; ξ

o = 50.00; δ = 0.0

α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

L2−BA

H1−BA

(a)

−1.1 −1.05 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7−3

−2.75

−2.5

−2.25

−2

−1.75

−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

0.25

0.5

log(ξ*)lo

g(|φ

−φ h|/|

φ|)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

L2−BA

H1−BA

(b)

−1.1 −1.05 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7−3

−2.75

−2.5

−2.25

−2

−1.75

−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

0.25

0.5

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

L2−BA

H1−BA

(c)

−1.1 −1.05 −1 −0.95 −0.9 −0.85 −0.8 −0.75 −0.7−3

−2.75

−2.5

−2.25

−2

−1.75

−1.5

−1.25

−1

−0.75

−0.5

−0.25

0

0.25

0.5

log(ξ*)

log(

|φ−

φ h|/|φ|

)


α1 = α

2 = 0

α1 = α

2 = 1

α1 = α

2 = (1/2)

α1 ≠ α

2

L2−BA

H1−BA

(d)

Figure 80: Convergence of the relative error in the l∞ Euclidean norm using β = (π/9) andRobin boundary conditions. The wavenumber ξo and the mesh distortion parame-ter used are: (a) ξo = 50, δ = 0 ; (b) ξo = 100, δ = 0 ; (c) ξo = 50, δ = 0.2 and (d)ξo = 100, δ = 0.2.

nique makes sense, i. e.linear interpolation on simplices and multilinear interpolationon blocks.

The α-interpolation of FEM and FDM on a rectangular domain discretized by struc-tured simplicial FE mesh would yield a scheme identical to the alpha-interpolationmethod (AIM) [110, 140] wherein the mass matrix that appears in the Galerkin FEMis replaced by an α-interpolated mass matrix. In the current PG method we recover


the AIM (even on unstructured simplicial meshes) by making the choice α1 = 0. Un-fortunately in this case the dispersion accuracy drops to second-order.

Recall that on square meshes many existing higher-order compact schemes (includ-ing the QSFEM [10]) can be recovered by an appropriate choice of the parametersα1,α2 [137]. As most of the expressions for α1,α2 optimized for square meshes neednot be optimal for unstructured meshes in the presented examples we have con-sidered only the simplest expressions that would guarantee fourth-order (choosingα1 = α2 = (1/2)) and sixth-order (α1,α2 given by Eq.(5.40b)) dispersion accuracyon square meshes. Convergence studies of the solution error corresponding to thesetwo choices are done to quantify the pollution effect and comparisons are made withrespect to the errors of the Galekin FEM, the nodally exact interpolant and the bestapproximations in the L2 norm and the H1 semi-norm respectively. Both the Dirichletand Robin boundary conditions were considered in the examples. The wavenumbersξo = 50 and ξo = 100 were chosen to represent values in the mid-frequency andhigh-frequency range respectively.

For the Dirichlet problem, the results on square meshes verify the higher-order dis-persion accuracy and the low pollution effect. However on unstructured meshes thedispersion accuracy of the current PG method drops down to second-order (verifiedby the errors in the l∞ Euclidean norm). Also, the performance of both the choices forthe parameters α1,α2 is similar on unstructured grids. For the mid-frequency range,i. e.ξo = 50 the errors in the l∞ Euclidean norm for both the parameter choices is closeto the error of the best approximation in the L2 norm. In the high-frequency range,i. e.ξo = 100, the improvement with respect to the Galerkin FEM is significant. How-ever, the solutions exhibit spurious modulations indicating that there is still room forimprovement.

For the Robin problem, these spurious modulations in the solutions cease to exist.The pollution effect on square meshes is greatly reduced and on unstructured meshesit is small. Also, the location of the error lines of the current PG method is in betweenthe error lines of pollution-free solutions, viz. the nodally-exact FE interpolant andthe best approximations in the L2 norm and the H1 semi-norm, thus indicating highaccuracy.

The additional cost of implementation of the current PG method is just the evalua-tion of the element boundary integrals. All the algebraic evaluations are done at theelement level unlike the QOPG method [128] where it is done at the patch level. Thisfeature allows the current PG method to be easily incorporated within an ‘assemble-by-elements’ data structure. The choice of the parameters α1 = α2 = (1/2) render thecurrent PG method independent of the problem and mesh data. In this sense and forthis choice, the current PG method could be labeled ‘parameter-free’.

Part III

S T O K E S P R O B L E M

O Lord, just as different rivers with distinct sources join the sea,the different paths which people take through different tendencies,various though they appear, crooked or straight, all lead to Thee!

— Hymn 7, Shiva Mahimnha Stotram.

6P R E S S U R E L A P L A C I A N S TA B I L I Z AT I O N

6.1 introduction

Many stabilization procedures for solving incompressible problems in fluid mechan-ics using the finite element method (FEM) have been proposed [1, 2, 11, 14, 22, 34–39, 42, 43, 52, 54, 56, 63, 90, 93, 96, 98, 141, 142, 171, 177–179, 195]. Earlier proce-dures were based on the so-called penalty approach. This method assumes a pseudo-compressible behavior for the flow with a relationship between the volumetric strainrate εv and the pressure p expressed as [54, 195]

εv =1

αp (6.1)

where α is a large number playing the role of an “artificial” bulk parameter for thefluid. Clearly for α→∞ the full incompressibility condition εv → 0 is recovered. An-other family of stabilized methods added to the standard incompressibility equation(either in the strong form or in the variational expression) a Laplacian of pressureterm scaled by a stabilization coefficient depending on physical parameters and thetime step increment. Some of these stabilization methods are described in [54, 195]. Asimilar stabilization procedure adds to the variational equation a local L2 polynomialpressure projection multiplied by the inverse of the kinematic viscosity [52]. Theseapproaches are inconsistent since the stabilization term does not vanish for the exactsolution, which can lead to errors in the pressure distribution and in the conservationof the total volume for some problems. An improved stabilization technique adds tothe incompressibility condition a term that is a function of the discretized momentumequations, thus ensuring consistency. A popular procedure of this kind is the GalerkinLeast Square (GLS) method [56, 93]. In the GLS procedure the stabilized variationalexpression for the incompressibility equation has the following form∫

Ω

qεvdΩ−

∫Ω

τ(∇Tq)rmdΩ+ boundary terms = 0 (6.2)

where Ω is the analysis domain, q are test functions, ∇ is the gradient operator, τis a stabilization parameter and rm is a vector containing the discrete residuals ofthe momentum equations written as rm = 0. The boundary terms in Eq.(6.2) areadded to ensure the consistency of the method [56]. The GLS method is an efficientand accurate stabilization procedure for incompressible flows provided the boundaryterms are properly accounted for in Eq.(6.2).

Another residual-based stabilization technique is the so-called pressure-gradient pro-jection (PGP) stabilization [35, 36]). In the PGP method, pressure gradients are pro-jected onto a continuous field and the difference between the actual gradients and

157

158 6 pressure laplacian stabilization

their own projections generates the stabilization terms. This is equivalent to replacingthe variational expression for the incompressibility equation by the following equa-tion ∫

Ω

qεvdΩ−

∫Ω

τ(∇Tq)(∇p+π) = 0 (6.3)

where π is a continuous function (termed the pressure-gradient projection vector)obtained by projecting the pressure gradient ∇p on the velocity field, and τ is astabilization parameter.

The term (∇p + π) in Eq.(6.3) can be interpreted as a residual term if we writethe momentum equations as rm = ∇p+ π. The total number of discrete unknownsis increased by the π field, which is discretized via pressure shape functions. Forcompleteness, the set of governing equations is extended with additional equationsrequiring the vanishing of the sum (∇p+ π) in a weighted residual sense. This pro-vides the equations for computing the π variables.

A variant of the PGP technique is the orthogonal sub-scales (OSS) method [11, 34, 39].The variational expression for the incompressibility constraint is written in the OSSmethod as∫

Ω

qεvdΩ−

∫Ω

τ(∇Tq)(rm +π)dΩ = 0 (6.4)

where rm is the discrete residual of the momentum equations and π are additionalvariables that are now interpreted as the projection of the momentum residuals intothe velocity space (without boundary conditions). The term rm + π represents an en-hanced approximation to the exact momentum residuals rm. Consistency is preservedby enforcing that the sum rm+π vanishes in a weighted residual sense. This also pro-vides the closure equations for computing the π variables.

PGP and OSS stabilization methods are useful for homogeneous flows lacking free-surfaces but encounter difficulties to satisfy incompressibility for fluids with hetero-geneous (and discontinuous) physical properties [45, 103, 104] and, in some cases,for free-surface flows when pressure segregation techniques are used for solving theNavier-Stokes equations. Furthermore, PGP and OSS methods increase the numberof problem variables (u,p and π) as well as the connectivity (bandwidth) of the stabi-lization matrices to be solved.

The stabilization parameter τ in the GLS, PGP and OSS methods is typically chosenas a function of the viscosity and the mesh size [1, 2, 11, 14, 22, 34–39, 42, 43, 52, 54,56, 63, 90, 93, 96, 98, 141, 142, 171, 177–179, 195]. However, the optimal definition ofthe stabilization parameter is still a challenge in these methods.

A pressure Laplacian stabilization (PLS) method was introduced in [158], that addstwo stabilization terms to the variational form of the incompressibility equation: (1) apressure Laplacian, and (2) a boundary integral. Both terms are multiplied by resid-ual dependent stabilization parameters which emerge naturally from the formulation.Consistency is preserved since the stabilization parameters vanish for the exact so-lution. The Laplace matrix and the boundary matrix are computed at element level.Because pressure gradient continuity is not enforced, as it happens, for instance, instandard PGP methods, the treatment of heterogeneous multi-fluid problems, such asmixing, is facilitated.

6.2 Governing equations 159

The aim of this chapter is to show that many of the stabilized methods describedin the previous lines, and some new ones, can be derived starting from the modifiedmass balance equation obtained via first and second order finite calculus (FIC) pro-cedures. The FIC technique is based on writing the balance equations in mechanicsin a domain of finite size and retaining higher order terms in the Taylor series expan-sions used for expressing the derivative field in the vicinity of a fixed point in thedomain. The resulting modified balance equation contains the traditional terms ofinfinitesimal theory plus additional terms that depend on the dimensions of the bal-ance domain and the derivatives of the infinitesimal balance equations [141]. Clearly,as the dimensions of the balance domain tend to zero the classical balance laws of me-chanics are recovered. The interest of the additional terms in the FIC expressions isthat they naturally lead to the stabilized numerical schemes (such as stabilized FEM)in fluid and solid mechanics without the need of introducing additional assumptions[101, 102, 105, 141–144, 148–151, 153, 156, 157]. The FIC approach therefore is pre-sented here as a parent procedure for deriving a family of old and new residual-basedstabilized methods for the analysis of Stokes flows.

An apparent drawback of some of the residual-based stabilized methods presentedin this chapter is that the resulting stabilized equation is nonlinear (due to the resid-ual dependence of the stabilization parameters) and this requires using an iterativesolution scheme. Preliminary results obtained for simple Stokes flow problems solvedwith the PLS and PGP methods show that the convergence of the PLS solution is typi-cally found in 2-3 iterations [158]. Also, the nonlinearity can be easily handled withina time integration scheme in transient problems, or in practical problems where othernon-linearities might appear due to the presence of convective terms in the momen-tum equations or non-linear material behavior. In the last part of this chapter, theperformance of the PLS method is compared with that of the GLS, PLS and OSS tech-niques for some relatively simple but illustrative examples of application to Stokesflow problems.

6.2 governing equations

The equations for an incompressible Stokes flow are expressed in the usual manneras:

Momentum

ρDviDt

−∂σij

∂xj− bi = 0 on Ω (6.5)

Mass balance (incompressibility)

εv :=∂vi∂xi

= 0 on Ω , i = 1, 2, 3 (6.6)

In Eqs.(6.5) and (6.6), Ω is the analysis domain with a boundary Γ , vi is the velocityalong the ith coordinate direction, ρ is the density, σij are the Cauchy stresses, bi arethe body forces (typically bi = ρgi where gi is the component of the gravity alongthe ith direction). In our work we assume that DviDt = ∂vi

∂t , i. e.convective derivativeterms are neglected, as it is usual in Stokes flows and Lagrangian descriptions ofincompressible continua [101, 102, 105, 149, 150, 157, 194].


The problem is completed with the boundary conditions for velocities and tractions,i. e.

vi − vpi = 0 on Γu (6.7a)

σijnj − tpi = 0 on Γt (6.7b)

where vpi denote the prescribed velocities on the Dirichlet boundary Γu and tpi arethe traction forces acting on the Neumann boundary Γt, with the normal vector n =

[n1,n2,n3]T (for 3d problems). The total boundary is Γ := Γu ∪ Γt.In Eqs.(6.5)–(6.7) and in the following, summation convention for repeated indices

in products and derivatives is used unless otherwise specified.Following standard practice, Cauchy stresses are split into deviatoric and pressure

components as

σij = sij + pδij (6.8)

where sij are deviatoric stresses, p = σii/3 is the pressure (assumed here to be positiveif the mean normal stress is tensile) and δij is the Kronecker delta.

We will also assume the constitutive equations of an isotropic Newtonian viscousfluid for which deviatoric stresses are related to deformation rates εij by

sij = 2µ

(εij −

1

3εvδij

)(6.9a)

where µ is the fluid viscosity and

εij =1

2

(∂vi∂xj

+∂vj

∂xi

), εv := εii (6.9b)

6.3 integral form of the momentum equations

The weighted residual form of Eqs.(6.5) and (6.7) is

∫Ω

wi

[ρ∂vi∂t

−∂σij

∂xj− bi

]dΩ+

∫Γt

wi(σijnj − tpi )dΓ = 0 (6.10)

where wi are components of an appropriate test function.Integrating by parts the term involving σij in Eq.(6.10) and substituting Eq.(6.8)

into the expression for σij gives an integral expression of the momentum equationsas ∫

Ω

[wiρ

∂vi∂t

+∂wi∂xj

sij +∂wi∂xi

p

]dΩ−

∫Ω

wibidΩ−

∫Γt

witpi dΓ = 0 (6.11)

Eq.(6.11) is the starting point for the finite element discretization of the momentumequations.

6.4 stabilized form of the incompressibility equation using finite

calculus

Here we present the finite calculus (FIC) technique and two stabilized forms for themass balance equation using first and second order FIC techniques.

6.4 Stabilized form of the incompressibility equation using finite calculus 161

6.4.1 Finite calculus

The finite calculus technique is based on writing the balance equations of mechanics ina domain of finite size and retaining higher order terms in the Taylor series expansionused for expressing the integrand (i. e.the classical balance equation in infinitesimaltheory) in the vicinity of a fixed point in the domain. The resulting modified balanceequation contains the traditional terms of infinitesimal theory plus additional termsthat depend on the dimensions of the balance domain and the derivatives of theinfinitesimal balance equations [141–143].

First we recall some standard identities/results which will be used to arrive at theFIC modified balance equation. Consider an arbitrary finite-size balance domain Ωbused to express the balance of fluxes/mass about an arbitrary point P (see Figure 81a).The centroid G of the domain Ωb is found as,

xG :=

∫Ωb

x dΩ∫Ωb

dΩ⇒∫Ωb

(x − xG) dΩ = 0 (6.12)

The moment of inertia of the domain Ωb is expressed as,

I :=∫Ωb

[(x · x)δ− (x⊗ x)] dΩ =

∫Ωb

(|x|2δij − xixj

)dΩ (6.13)

Clearly, I is real, symmetric and has a full rank. The spectral theorem for tensorsguarantee the existence of principle axes for I. By orienting the reference frame alongthe principle axes, the products of inertia become zero, i. e.Iij = 0 ∀i 6= j. In otherwords, the tensor I is diagonalized in this configuration.

P

Ωb

(a)

P

G

Ωb

ℓ

(b)

P = G, ℓ = 0

Ωb

(c)

Figure 81: (a) An arbitrary finite-size balance domain Ωb used in FIC to express the balanceof fluxes/mass about an arbitrary point P. (b) The centroid G and the principleaxes of Ωb are now shown. The distance vector from P to G is represented by `. (c)The balance domain Ωb is translated such that P = G and is rotated about P suchthat its principle axes are aligned with the reference axes. In this configuration theproducts of inertia of Ωb are zero.

The balance laws of mechanics in the domain Ωb can be expressed as follows,∫Ωb

R(x) dΩ = 0 (6.14)


where R(x) = 0 is the classical balance equation in infinitesimal theory. Using themultidimensional Taylor series expansion, the residual R(x) can be expressed withrespect to the point P as follows,

R(x) = R(xP) + x ·∇R(xP) + 12

x ·HR(xP) · x + h.o.t (6.15a)

x := x − xP (6.15b)

where H(·) and h.o.t represent the Hessian operator and the higher-order terms re-spectively. Substituting Eq.(6.15a) in Eq.(6.14) we have,∫

Ωb

[R(xP) + x ·∇R(xP) + 1

2x ·HR(xP) · x + h.o.t

]dΩ = 0 (6.16a)

⇒ R(xP)∫Ωb

dΩ+∇R(xP) ·∫Ωb

x dΩ+ HR(xP) :∫Ωb

1

2(x⊗ x) dΩ+ h.o.t = 0

(6.16b)

Using Eq.(6.12) in Eq.(6.16b) we get,

R(xP) + ` ·∇R(xP) + L : HR(xP) + h.o.t = 0 (6.17a)

` := (xG − xP) ; L :=

∫Ωb

(x⊗ x) dΩ

2∫Ωb

dΩ(6.17b)

Now as this point P can be arbitrarily chosen, the FIC-modified balance equationcan be written after dropping out the higher-order terms (h.o.t) as follows:

R(x) + ` ·∇R(x) + L : HR(x) = 0 (6.18)

As the finite balance-domain Ωb may be arbitrarily chosen for a given point, gener-ally ` and L may vary arbitrarily from one point to the other. However we may chooseto fix the shape of Ωb for all the points that belong to a particular sub-domain of theproblem. For instance consider a partition of the problem domain into finite elements.Then for each element K we may fix some shape for Ωb and hence obtain a constantvalue for ` and L within K. We may also choose a fixed shape of Ωb for the entireproblem domain and thus obtaining a constant ` and L everywhere. Henceforth wewill only consider the choice of Ωb to be fixed for any given element K but may varyfrom one element to the other.

Remark: Choosing the shape for Ωb and the location of the sampling point P withinΩb defines the FIC model and the associated FIC-modified balance equation.

Choosing Ωb such that its centroid G coincides with P, we obtain ` := xG − xP = 0

(see Figure 81b). In this FIC model, the first order terms in the FIC-modified residualvanishes. So we have,

R(x) + L : HR(x) = 0 (6.19)

Note that the matrix L defined earlier in Eq.(6.17b) has full rank and it can be diag-onalized by orienting Ωb such that its principle axes are aligned with the coordinateaxes (see Figure 81c). This is directly related to the fact that the off-diagonal elementsof L and moment of inertia I of Ωb have the same expressions (except for a change ofsign).

6.4 Stabilized form of the incompressibility equation using finite calculus 163

6.4.2 First order FIC form of the incompressibility equation

The first order FIC form for the incompressibility equation (R(x) := εv) is found byretaining only the first order derivatives in the FIC-modified balance equation givenby Eq.(6.18). To fix ideas we choose the balance domain Ωb to be rectangular and thesampling point P situated at either the top-right or bottom-left corner. The resultingexpression is

εv ±1

2hT∇εv = 0 (6.20)

For 2d problems h = [h1,h2]T where h1,h2 are the dimension of the rectangulardomain Ωb. The sign in Eq.(6.20) is positive or negative depending on whether thesampling point P is the bottom-left or the top-right corner node of Ωb respectively.The sign in Eq.(6.20) is irrelevant in practice. The original derivation of Eq.(6.20) froma different point-of-view can be found in [141].

6.4.3 Higher order FIC form of the incompressibility equation

The higher order FIC incompressibility equation is found by using the FIC model thatgives the FIC-modified balance equation given by Eq.(6.19). As discussed earlier onecan arrive at this form by choosing the balance domain Ωb such that its centroid Gcoincides with the sampling point P (see Figure 81). Again, to fix ideas we choose arectangular balance domain Ωb and the sampling point P situated at its center. Forthis choice we have,

` = 0 , L =1

24

[h21 0

0 h22

](6.21)

Thus for 2d problems the higher order FIC incompressibility equation is written as,

εv +h2124

∂2εv

∂x21+h2224

∂2εv

∂x22= 0 (6.22)

The original derivation of Eq.(6.22) from a different point-of-view is shown in [158].Clearly for the infinitesimal case h1 = h2 = 0, the standard incompressibility equation(εv = 0) is recovered for both Eqs.(6.20) and (6.22).

Eqs.(6.20) and (6.22) can be interpreted as non-local mass balance equations incor-porating the size of the domain used to enforce the mass balance condition and spacederivatives of the volumetric strain rate. The FIC mass balance equations can be ex-tended to account for temporal stabilization terms. These terms, however, are disre-garded here as they have not been found to be relevant for the problems investigatedso far.


6.5 on the proportionality between the pressure and the volumet-ric strain rate

Let us assume a relationship between the pressure and the volumetric strain ratetypical for “compressible” and “quasi-incompressible” fluids, as

1

Kp = εv (6.23)

where K is the bulk modulus. Clearly for a fully incompressible fluid K = ∞ andεv = 0. For finite, although very large, values of K the following expression is readilydeduced from Eq.(6.23)

1

K∇p = ∇εv (6.24)

where ∇ is the gradient operator. For 2d problems, ∇ =[∂∂x1

, ∂∂x2

]T.

Eq.(6.24) shows that pressure and volumetric strain rate gradients are co-directionalfor any K 6= 0. We will assume that this property also holds for the full incompressiblecase (at least for values of K comfortably representable on the computer withoutoverflow). From Eqs.(6.23) and (6.24) we deduce

∇εv|∇εv|

=∇p|∇p|

,1

εv

∂εv

∂xi=1

p

∂p

∂xi(6.25a)

and hence

∂εv

∂xi=εv

p

∂p

∂xi=

|∇εv||∇p|

∂p

∂xi(6.25b)

6.6 a penalty-type stabilized formulation

The first order FIC mass balance equation (6.20) is written as (taking the negative sign)

εv =hj

2

∂εv

∂xj(6.26)

Using Eq.(6.25b), the FIC term in the r.h.s. of Eq.(6.26) can be expressed as follows

εv =

(hj

2

εv

p2∂p

∂xj

)p =

1

αp (6.27)

where

α =2p2

hjεv

(∂p

∂xj

)−1

(6.28)

is a pressure stabilization parameter [54, 194, 195].Eq.(6.27) is equivalent to the so-called penalty formulation (see Eq.(6.1)) for which

the pressure-volumetric strain rate relationship is expressed as p = αεv where α apenalty parameter that plays the role of a large “artificial” bulk modulus.

6.7 Galerkin-least squares (GLS) formulation 165

Clearly, α→∞ for values of εv → 0. However, the full incompressibility condition(εv = 0) at element level is obtained on rare occasions only. Hence, the form for α ofEq.(6.28) provides a consistent (residual-based) definition for the penalty stabilizationparameter. Nevertheless, upper and lower cut-off values for the values for α shouldbe imposed to prevent volumetric locking when εv or ∂p

∂xiare equal to zero or the

vanishing of α in zones where p is zero [54, 194, 195].The weighed residual form of Eq.(6.27) is∫

Ω

q(εv −1

αp)dΩ = 0 (6.29)

where q are adequate test functions.

6.7 galerkin-least squares (gls) formulation

The starting point is now the higher order FIC incompressibility equation (6.22). Theweighted residual form of this equation is∫

Ω

q

(εv +

h2i24

∂2εv

∂x2i

)dΩ = 0 (6.30)

Integration by parts of the second term in Eq.(6.30) gives (for 2d problems)

∫Ω

qεvdΩ−

∫Ω

(2∑i=1

h2i24

∂q

∂xi

∂εv

∂xi

)dΩ+

∫Γ

q

24

(2∑i=1

nih2i

∂εv

∂xi

)dΓ = 0 (6.31)

where ni are the components of the unit normal vector to the boundary Γ .In the derivation of Eq.(6.31), space derivatives of the characteristic lengths h1 and

h2 have been neglected. This is correct if we assume that the value of the characteristiclengths is fixed at each point in space. In any case, this assumption does not invalidatethe derivation, as long as the discretized formulation converges to correct velocityand pressure fields satisfying the momentum and incompressibility equations in aweighted residual sense and up to the desired order of accuracy.

The term ∂εv∂xi

in Eq.(6.31) is expressed as follows. The momentum equations (6.5)can be written using Eqs.(6.8) and (6.9a)

ρ∂vi∂t

−∂

∂xj

(2µεij

)+2

3µ∂εv

∂xi−∂p

∂xi− bi = 0 (6.32)

From Eq.(6.32) we deduce (neglecting space variations of the viscosity)

∂εv

∂xi=3

2µrmi

and ∇εv =3

2µrm (6.33)

where

rmi:= −ρ

∂vi∂t

+∂

∂xj

(2µεij

)+∂p

∂xi+ bi (6.34a)

is the form of the momentum residuals used in the subsequent derivations. Note thatrmi

= 0 for the exact incompressible solution.


For 2d problems

rm = [rm1, rm2

]T (6.34b)

Substituting ∂εv/∂xi from Eq.(6.33) into (6.31) gives∫Ω

qεvdΩ−

∫Ω

(2∑i=1

τi∂q

∂xirmi

)dΩ+

∫Γ

q

(2∑i=1

τinirmi

)dΓ = 0 (6.35)

with

τi =h2i16µ

(6.36)

The form of Eq.(6.35) is equivalent to that obtained in the Galerkin Least Square(GLS) formulation [93] with the boundary integral modification presented in [56].The expression for the stabilization τi of Eq.(6.36) is similar to that typically found inthe stabilization literature for Stokes flows [54, 56, 93, 195].

Expansion of the residual term within the second integral yields the standard Lapla-cian of pressure plus additional terms, i. e.

∫Ω

qεvdΩ−

∫Ω

(2∑i=1

τi∂q

∂xi

∂p

∂xi

)dΩ

−

∫Ω

2∑i=1

(τi∂q

∂xi

[− ρ

∂vi∂t

+∂

∂xj(2µεij) + bi

])dΩ

+

∫Γ

q

(2∑i=1

τini

[− ρ

∂vi∂t

+∂p

∂xj+

∂

∂xi(2µεij) + bi

])dΓ = 0 (6.37)

Clearly for linear FE approximations the viscous terms vanish in Eq.(6.37).We note that the FIC approach presented here introduces the GLS-type stabilization

terms just in the incompressibility equation. This is a difference with the standardGLS method which also introduces stabilization terms in the momentum equations[93]. These terms provide symmetry of the global system of equations and are usefulfor analysis of Navier-Stokes flows. However, they are typically unnecessary for theanalysis of Stokes flows. An exception is some transient problems when small timesteps are used [90].

6.8 pressure laplacian stabilization (pls) method

6.8.1 Variational form of the mass balance equation in the PLS method

Using the relationships in Eq.(6.25b) we can write the second integral in Eq.(6.31) as∫Ω

(2∑i=1

h2i24

∂q

∂xi

∂εv

∂xi

)dΩ =

∫Ω

(2∑i=1

h2i∂q

∂xi

∂p

∂xi

)|∇εv|24|∇p|dΩ

=

∫Ω

(2∑i=1

τi∂q

∂xi

∂p

∂xi

)dΩ (6.38)

6.8 Pressure Laplacian stabilization (PLS) method 167

with the stabilization parameters given by

τi =h2i |∇εv|24|∇p| , i = 1, 2 (for 2d problems) (6.39)

Substituting Eqs.(6.38) into (6.31) we write the stabilized mass balance equation as∫Ω

qεvdΩ−

∫Ω

(∇Tq)Dv∇pdΩ+

∫Γ

qgdΓ = 0 (6.40)

For 2d problems

Dv =

[τ1 0

0 τ2

]and g =

2∑i=1

h2i24ni∂εv

∂xi(6.41)

where Dv is a matrix of stabilization parameters.

6.8.2 Computation of the stabilization parameters in the PLS method

From Eq.(6.33) we deduce (neglecting space variations of the viscosity)

2

3µ|∇εv| = |rm| (6.42)

From the first order FIC mass balance equation (6.20) (with the negative sign) wededuce

1

2hξ|∇εv| = εv (6.43)

where hξ is the projection of h along the gradient of εv, i. e.

hξ =hi

|∇εv|∂εv

∂xi=

hT∇εv|∇εv|

(6.44)

Eqs.(6.42) and (6.43) are consistently modified as follows

2

3µ|v||∇εv| = |v||rm| (6.45)

1

2phξ|∇εv| = pεv (6.46)

In Eq.(6.45) v is the velocity vector.The r.h.s. of Eqs.(6.45) and (6.46) represents the power of the residual forces in the

momentum equations and of the volumetric strain rate, respectively. Note that theproduct phξ in Eq.(6.46) is always positive, as pεv > 0 (see Eq.(6.23)). Positiveness ofpεv pointwise is however ensured in the computation by taking the modulus of thisproduct in the subsequent expressions.

From Eqs.(6.45) and (6.46) we deduce

|∇εv| =|pεv|+ |v||rm|12 |phξ|+

23µ|v|

(6.47)


Substituting Eq.(6.47) into (6.39) gives the expression for the stabilization parame-ters as

τi =h2i (|pεv|+ |v||rm|)

(12|phξ|+ 16µ|v|)|∇p|(6.48)

The expression for τi in Eq.(6.48) will vanish for values of vi and p satisfying ex-actly the incompressibility equation (εv = 0) and the momentum equations (rm = 0).Clearly for the discrete problem, the stabilization parameters depend on the numer-ical errors in the approximation for εv and rm. In practice, it is advisable to choosea cut-off value for the lower and upper bounds for τi avoiding very small or toolarge values of the stabilization parameter (for instance in zones where ∇p is small).In the examples shown in this chapter we have chosen the following limiting band:10−8 6 τi 6 105.

remark 1 . Using just Eq.(6.42) for defining |∇εv| and substituting this into Eq.(6.39)gives

τi =h2i |rm|

16µ|∇p|(6.49)

In absence of body forces and assuming steady state conditions and a linear FEapproximation, then |rm| = |∇p| and

τi =h2i16µ

(6.50)

which coincides with Eq.(6.36) deduced for the GLS method. The dimension ofτi is s×m

3

kg . The expression for τi in Eq.(6.50) is typically found in the stabilizedFEM literature for Stokes flow [1, 2, 14, 36, 54, 90, 141, 142, 177, 195].

remark 2 . Other residual-based expressions for the stabilization parameters τi inthe PLS method can be found. For instance, two alternative expressions for τiare

τi =h2i|∇p|

(ρ|v|+ ρ|hξ|

2∆t

)|εv|+ |rm|

(12ρ|hξv|+ 6ρ

h2ξ∆t + 16µ

) (6.51a)

and

τi = h2i

ρ|hξv|+ µ

24ρ|hξv|| pεv |+ 16µ2 |∇p|

|rm|

(6.51b)

The derivation of above expressions can be found in [158].

6.8.3 PLS boundary stabilization term

From the relationships in Eq.(6.25b) we express the boundary term g of Eq.(6.41) as

g =

2∑i=1

h2i24ni∂εv

∂xi=

2∑i=1

h2i24ni

|∇εv||∇p|

∂p

∂xi=

2∑i=1

τini∂p

∂xi(6.52)

6.9 Pressure-gradient projection (PGP) formulation 169

The boundary integral in Eq.(6.40) can therefore be expressed in terms of the pres-sure gradient components using Eq.(6.52) as

∫Γ

q

(2∑i=1

τini∂p

∂xi

)dΓ (6.53)

where all the terms within the integral are computed at the boundary Γ .

remark 3 . For hi = hj = h then τi = τ. In this case, the boundary integral (6.40)can be expressed as∫

Γ

qτ∂p

∂ndΓ (6.54)

where ∂p∂n = ni

∂p∂xi

is the gradient of the pressure along the direction normal tothe boundary.

We note that all the expressions for τi given in the previous equations are “solutiondependent”. The nonlinear definition of the stabilization parameters can be useful forovercoming the limitations of the standard definitions of τi. A recent evidence of theusefulness of solution-dependent stabilization parameters can be found in [90].

6.9 pressure-gradient projection (pgp) formulation

An alternative stabilized formulation can be derived from the higher order FIC equa-tions by introducing the so-called pressure-gradient projection variables. The resultingstabilized mass balance equations can be derived is a number of ways [35, 36, 54,142, 195]. Here we show how a PGP (for pressure-gradient projection) method can bereadily obtained following the higher order FIC approach.

From the momentum equations it can be found

h2i24

∂εv

∂xi= τirmi

(6.55)

where rmiis defined in Eq.(6.34a). An expression for the stabilization parameter τi is

[35, 36, 54, 142, 195]

τi =h2i24

[ρl2

4∆t+2µ

3

]−1(6.56)

and l is a typical grid distance. The expression for τi of Eq.(6.56) can be obtained as aparticular case of Eq.(6.51a).

For relatively fine grids the numerical solution is insensitive to the values of hi :=αil [148]. In [158] we found that good results are obtained for the range of values ofhi such that

√2 6 αi 6

√6, where l is a typical grid distance.

For the steady state problems solved in this work the form of τi of Eq.(6.36) is usedwith αi = α =

√6.

Note that the stabilization parameters in the PGP method are constant for eachelement. This is an important difference versus the PLS and penalty formulations


of previous sections, where a nonlinear (and consistent) form for the stabilizationparameters is used.

In the standard PGP method the momentum residuals rmiare split as rmi

:= ∂p∂xi

+

πi where πi are the so-called pressure-gradient projection variables [35, 36]. In ourwork we use a slight different approach and split the momentum equations as

rmi:=

∂p

∂xi+1

τiπi (6.57)

where

πi = τi

(−ρ∂vi∂t

+∂sij

∂xj+ bi

)(6.58)

is the ith pressure-gradient projection weighted by the ith stabilization parameter. Theπi’s are now taken as additional variables which are discretized with the standardFEM in the same manner as for the pressure.

The split of Eq.(6.57) ensures that the term 1τiπi is discontinuous between adjacent

elements after discretization. This is essential for accurately capturing high discontin-uous pressure gradient jumps typical of fluids with heterogeneous physical proper-ties (either the viscosity or the pressure) [45, 104, 105]. In this manner the term 1

τiπi

can match the discrete pressure gradient term ∂p∂xi

which is naturally discontinuousbetween elements for a linear approximation of the pressure.

Substituting Eq.(6.55) into the second and third integral of Eq.(6.31) and using (6.57)gives (for 2d problems)∫

Ω

qεvdΩ−

∫Ω

2∑i=1

∂q

∂xi

(τi∂p

∂xi+ πi

)dΩ+

∫Γ

q

2∑i=1

ni

(τi∂p

∂xi+ πi

)dΓ = 0 (6.59)

The boundary integral in Eq.(6.59) is typically neglected in PGP formulations andwill be disregarded from here onward.

The following additional equations are introduced for computing the pressure gra-dient projection variables πi∫

Ω

2∑i=1

wi

(∂p

∂xi+1

τiπi

)dΩ = 0 (6.60)

where wi = q is usually taken.Recall that the term ∂p

∂xi+ 1τiπi (no sum in i) is an alternative expression for the

momentum residual equation (see Eq.(6.57)). This term is enforced to vanish in anaverage sense via Eq.(6.60).

6.10 orthogonal sub-scales (oss) formulation

The OSS method can be readily derived by writing the momentum residuals as

rmi= (rmi

+ πi) (6.61)

where rmiare the discrete residuals of the momentum equations and πi are additional

variables that are now interpreted as the projection of the discrete momentum residu-als into the velocity space (without boundary conditions) [11, 34, 39].

6.11 PLS+π method 171

The variational form for the incompressibility equations is obtained by substitutingEq.(6.61) into (6.35). This gives (neglecting the boundary terms)

∫Ω

qεvdΩ−

∫Ω

2∑i=1

∂q

∂xiτi(rmi

+ πi)dΩ = 0 (6.62)

The πi variables are computed by the following additional equations

∫Ω

2∑i=1

wiτi(rmi+ πi)dΩ = 0 (6.63)

with wi = q. Eq.(6.63) enforces the consistency of the method in an average sense.The stabilization parameters τi are computed as in the PGP method of previous

section.

remark 4 . . For linear FE interpolations the term rmiis simply

rmi= −ρ

∂vi∂t

+∂p

∂xi+ bi (6.64)

where vi and p are the approximate FE values for the velocities and the pressure.

6.11 pls+π method

The PLS method presented in §6.8 can be substantially enhanced if the momentumresiduals appearing in the expression for the stabilization parameters τi are computedusing Eq.(6.61). The resulting expression for τi is

τi =h2i (|pεv|+ |v||rm +π|)

(12|phξ|+ 16µ|v|) |∇p|(6.65)

This π variables are computed via Eq.(6.63) following the discretization proceduredescribed in the next section.

The expression for τi of Eq.(6.65) increases the efficiency and accuracy of the PLSmethod (see Example 6.15.5).

6.12 finite element discretization

6.12.1 Discretized equations

The domain Ω is discretized with a mesh of triangles or quadrilaterals (for 2d) andtetrahedra or hexahedra (for 3d).

For the penalty, GLS and PLS formulations, the velocities and the pressure areinterpolated over each element using the same approximation as (for 3d problems)

v =

v1

v2

v3

=

n∑j=1

Njvj , p =

n∑j=1

Njpj (6.66)


For the PGP and OSS formulations the πi variables are also interpolated using theshape functions Ni as

π =

π1

π2

π3

=

n∑j=1

Njπj (6.67)

In the above equations Nj = NjI3, Nj is the standard shape function for node j,I3 is the 3× 3 unit matrix, n is the number of nodes in the element (i. e.n = 3/4 forlinear triangles/tetrahedra) and vj, pj and πj are the values of the velocity vector,the pressure and the π variables vector at node j, respectively. Indeed, any other FEapproximation for v, p and π can be used.

Substituting the approximations (6.66) and (6.67) into the weak form of the mo-mentum equations (Eq.(6.10)) and in the adequate variational expression for the in-compressibility equation, gives the following global system of equations (for all themethods considered in this chapter)

M ˙a + Ha = f (6.68)

where ˙a = ddt a and the different matrices and vectors for the different stabilized FE

methods are

Penalty, GLS, PLS methods

a =

v

p

, M =

[M 0

M 0

], H =

[K Q

(QT + R) S

], f =

fvfp

(6.69a)

where

Penalty method : S = P, R = M = 0, fp = 0

GLS method : S = −L + B

PLS method : S = −L + B, R = M = 0, fp = 0

(6.69b)

PGP method

a =

v

p

π

, M =

M 0 0

0 0 0

0 0 0

, H =

K Q 0

QT −L −C

0 −CT −T

, f =

fv0

0

(6.70)

OSS method

a and H as in Eq.(6.70)

M =

M 0 0

Mp 0 0

Mπ 0 0

, f =

fvfpfπ

with fπ =

fπ1fπ2fπ3

(6.71)

The matrices and vectors in Eqs.(6.69)–(6.71) are formed by assembling the elementcontributions given in Box 1 for 3d problems.

6.12 Finite element discretization 173

Meij =

∫ΩeρNiNjdΩ, Keij =

∫Ωe

GTi DGjdΩ, Qeij =∫Ωe

GTi mNjdΩ

with

Gi = GNi with G =

∂

∂x10 0

∂

∂x2

∂

∂x30

0∂

∂x20

∂

∂x10

∂

∂x3

0 0∂

∂x30

∂

∂x1

∂

∂x2

T

m =1 1 1 0 0 0

, D = µ

[2I3 0

0 I3

]

Peij =

∫Ω

1

αNiNjdΩ, Leij =

∫Ωeτk∂Ni∂xk

∂Nj

∂xkdΩ, Beij =

∫Γ

3∑k=1

(τknkNi

∂Nj

∂xk

)dΓ

Meij =

∫Ωe

mTGiρ[τ]NjdΩ−

∫ΓeNiρτ

TnNjdΓ

(Meπ)ij = −

∫ΩeρNTi [τ]NjdΩ

Mep as Me neglecting the boundary term.

Reij = −

∫Ωe

(∇TNi)[τ]GT (DGj)dΩ+

∫ΓeNiτ

TnGT (DGj))dΓ

τ = [τ1, τ2, τ3]T , [τ] =

τ1 0 0

0 τ2 0

0 0 τ3

, τn = [τ]n

PGP : Ceij =∫Ωe

mTGiNjdΩ, Teij =∫Ωe

Ni[τ]−1NjdΩ

OSS : Ceij =∫Ωe

mTGi[τ]NjdΩ, Teij =∫Ωe

Ni[τ]NjdΩ

fevi =∫ΩeNibdΩ+

∫Γet

NitpdΓ , i, j = 1, 2, 3

fepi =

∫Ωe

3∑j=1

τj∂Ni∂xj

bj

dΩ−

∫ΓeNi

3∑j=1

τjnjbj

dΓ , (fπj)i = −

∫Ω

NiτjbdΩ

fpi as fpi neglecting the boundary term.I3 : 3× 3 unit matrix , b = [b1,b2,b3]T , tp = [tp1 , tp2 , tp3 ]

T

Γet : boundary of element e coincident with the external Neumann boundary

Box 2: Element expressions for the matrices and vectors in Eqs.(6.69)–(6.71) for 3d problems.The expression for τi changes for the different stabilized methods


remark 5 . For linear triangles, matrices M, P and T are computed with a three pointGauss quadrature. The rest of the matrices and vectors in Box 1 are computedwith just a one-point quadrature. A higher order quadrature might be requiredin some cases for integrating more accurately the nonlinear terms involving thestabilization parameters τi.

remark 6 . The contribution of the stabilization terms to the stiffness matrix K whichare typical in the standard GLS formulation are not taken into account in thiswork. These terms are irrelevant for the analysis of the steady-state Stokes prob-lems presented in this chapter.

6.12.2 Solution schemes for penalty, GLS, PLS methods

For penalty, GLS, PLS and combined methods, a monolithic transient solution ofEq.(6.68) can be found using the following iterative scheme

j+1an+1 =[jHn+1

]−1(

f +1

∆tMan

)(6.72a)

with

H = H +1

∆tM (6.72b)

In Eq.(6.72a) (·)n and (·)n+1 denote values at times t and t+∆t, respectively whilethe upper left index j denotes the iteration number; i. e.j(·)n+1 denotes values at timet+∆t and the jth iteration.

For the steady state problems solved in this work we have found the velocity andpressure variables simultaneously by inverting the system

Ha = f (6.73)

Clearly, for the PLS method the solution of Eq.(6.73) must be found iteratively, asthe stabilization parameters are a function of the velocity and the pressure. A simpledirect iteration scheme gives

j+1a = [jH]−1f (6.74)

6.12.3 Solution scheme for PGP and OSS methods

The solution of Eqs.(6.68) for the PGP and OSS methods is typically performed via aniterative staggered scheme.

The π nodal variables can be eliminated from Eqs.(6.68) as follows

PGP method : π = −T−1CT p (6.75a)

OSS method : π = −T−1(Mπ ˙v + CT p − fπ

)(6.75b)

6.13 Definition of the characteristic lengths 175

Substituting π from Eqs.(6.75) into the second row of Eqs.(6.68) yields the followingsystem of two equations for v and p

PGP method :M ˙v + Kv + Qp = fvQT v − (L − L)p = 0

(6.76a)

OSS method :M ˙v + Kv + Qp = fv(Mp − CT−1Mπ) ˙v + QT v − (L − L)p = fp − T−1fπ

(6.76b)

In Eqs.(6.76) L = CT−1CT is the discrete pressure Laplace matrix. This matrix has awider bandwidth than the Laplace pressure matrix L in Eqs.(6.69b). The differencebetween L and L provides the necessary stabilization for the accurate solution ofEqs.(6.76).

For the steady state case the solution for the velocity and pressure is found simul-taneously by Eq.(6.73) with

H =

[K Q

QT (L − L)

], a =

v

p

, f =

fvfp

(6.77)

where fp = 0 for the PGP method and fp = fp − T−1fπ for the OSS method.

remark 7 . The elimination of the π variables via Eq.(6.75) can be simplified by usinga diagonal form of matrix T obtained as Td = diag(T). This, however, does notaffect the bandwidth of matrix L.

remark 8 . The matrix multiplying ˙v in the second part of Eq.(6.76b) vanishes forthe case of constant density.

6.13 definition of the characteristic lengths

In §6.4.3 we had chosen a rectangular balance domain with the sampling point locatedat its centroid to arrive at the higher order FIC-modified incompressibility equation.Clearly, the expression for the tensor L obtained with this choice is anisotropic. In thiscase, we have to associate the definition of the balance domain to a procedure thatguarantees the objectivity of the resulting FIC-based method.

Unlike for the convection–diffusion–reaction problem where an anisotropic defini-tion of the characteristic length tensors was shown to provide better solutions (seeChapter 3), preliminary numerical experiments with the Stokes problem suggest oth-erwise. That is, no gain is obtained with respect to the quality of the solution (bothv and p) by defining L in an anisotropic manner. It is for this reason that we havechosen a constant value for hi defined (for 2d problems) as

hi = he with he = [ηAe]1/2 (6.78)

whereAe is the element area, η = 1 for multilinear elements and η = 2 for simplicialelements. This corresonds to the choice of a circular FIC balance domian with thesampling point located at its centroid and with radius h/

√6. So we have,

` = 0 , L =1

24

[h2 0

0 h2

](6.79)


6.14 some comments on the different stabilization methods

6.14.1 Penalty method

For the penalty method the system matrix H is symmetrical. Matrix M becomes ill-conditioned for very large values of the penalty parameter α. A cut-off for α whenthe solution approaches the incompressibility limit is therefore mandatory to avoidthe volumetric locking problem [54, 195].

6.14.2 GLS method

For the GLS method presented here matrices M, R and B are non symmetrical. Sym-metry of matrices M and H can be simply found by shifting the non symmetricalterms to the r.h.s. of Eqs.(6.68). This introduces a nonlinearity in the solution scheme.

6.14.3 PLS method

For the PLS method the boundary stabilization matrix B is non-symmetrical. Symme-try of the system matrix H can be recovered by shifting the boundary terms to ther.h.s. of Eq.(6.68). This gives

H =

[K Q

QT −L

], f =

fvqp

(6.80a)

For 3d problems

qepi =

∫ΓeNi

3∑j=1

τjnj∂p

∂xj

dΓ (6.80b)

The boundary force vector qp is computed at each iteration as part of the iterativesolution process. Preliminary experiences in applying this method show that this doesnot increase the total number of iterations.

6.14.4 PGP and OSS method

All matrices in the PGP method are symmetrical. For the OSS method matrix M is notsymmetrical. Symmetry of matrix M can be found by shifting the non symmetricalterms (involving time derivatives of the velocities) to the r.h.s. of Eq.(6.68).

6.14.5 Comparison between the different stabilization methods

1. In the PGP and OSS methods the stabilization parameter is typically taken asconstant (at least for homogeneous meshes and constant viscous fluids). In thePLS method, however, the stabilization parameter τi varies as a function of thevolumetric strain rate and the residual of the momentum equations.


2. In the PGP and OSS methods the amount of stabilization is variable in space.This variation is introduced by the difference between the Laplace pressure ma-trix L and the discrete pressure Laplace matrix L. In the PLS method (and also inthe penalty method presented here) the amount of stabilization is also variablein space, but the variation is introduced by the consistent stabilization parame-ters.

3. The consistency in the GLS methods is guaranteed by introducing the discreteresidual of the momentum equations in the stabilized mass balance equation.Consistency in the PGP and OSS methods is enforced by introducing additionalequations representing the vanishing of the momentum residuals in an averagesense. In the PLS method the consistency is guaranteed by the expression of thestabilization parameters which also vanish for the exact solution (i. e.for εv = 0

and rmi= 0).

4. The PLS method is nonlinear due to the dependence of the stabilization param-eters with the volumetric strain rate, the pressure, the pressure gradient and theresidual of the momentum equations. The PGP and OSS methods are non lineardue to the definition of the πi variables which are a function of the pressurefield. The GLS method can be considered as a linear method.

5. All methods, except the penalty method, introduce a boundary stabilizationterm. Accounting for this term is relevant in the GLS and PLS methods whenlower order finite elements are used. On the contrary, consistency recovery tech-niques as proposed in [111] are required. Unlike for the GLS method, it is ob-served that this dependence on such consistency recovery techniques is weak forthe PLS method. In other words, the adverse effect of excluding the boundarystabilization term is found to be small for the PLS method (see Examples 6.15.3and 6.15.4).

6. In PGP and OSS methods the boundary stabilization term is usually neglected.This simplification is acceptable on external boundaries, but cannot be neglectedat internal interfaces with a jump in the physical properties.

7. PGP and OSS methods typically yield identical results for problems when theforce term bi belongs to the space of finite element functions and linear (orbilinear) elements are used [35].

6.15 examples of application

We present a number of examples of simple steady-state Stokes flow problems. Theaim is to validate and compare the accuracy and efficiency of some of the methodspresented in this chapter. The methods compared are:

• PLS method of §6.8 using the expression for τi of Eq.(6.48). The effect of includ-ing or not the boundary integral (BI) terms of Eq.(6.40) has been studied.

• PLS+π method of §6.11. This method was used just in the manufactured flowproblem (Example 6.15.5).


• GLS method of §6.7, including and excluding the boundary integral (BI) termsof Eq.(6.37).

• OSS method of §6.10 with consistent and diagonal forms of matrix T.

The problems solved are the following:

i) Hydrostatic flow problem for a single fluid in a square domain.

ii) Two-fluid hydrostatic problem in a square domain.

iii) Poiseuille flow in a trapezoidal domain.

iv) Lid driven cavity flow problem.

v) Manufactured flow problem in a trapezoidal domain.

For all problems the nodal velocities and pressures have been found simultaneouslyunder steady-state conditions by solving Eq.(6.73). For the GLS and the OSS methods thesolution is found in a single step. For the PLS method the direct iteration scheme ofEq.(6.74) is used. The first PLS (and PLS+π) solution is found in all cases using aconstant value of τi = τ = 10−5. This roughly corresponds to the value of τi for theGLS and OSS methods given by Eq.(6.36).

The convergence of the nonlinear iterations for the PLS method is measured in theEuclidean vector norm for velocities and pressure measured as

‖jφ− j−1φ‖l2‖jφ‖l2

6 ε with φk =

v1k

v2k

pk

and ‖φ‖l2 =[∑a

(φa)2

]1/2

In our work we have chosen ε = 10−4 for examples i)–iv) and ε = 10−3 for examplev). A comparison of the PLS and PGP methods for problems i, ii and iv is reported in[158].

6.15.1 Hydrostatic flow problem for a single fluid in a square domain

We solve for the pressure distribution in a square container filled with water. Thebody forces are b1 = b2 = ρg with values of the density and gravity constant equalto ρ = 1000 Kg/m3 and g = −10 m/s2, respectively. The viscosity is µ = 10−3

Ns/m2. The normal velocity has been prescribed to zero at the bottom line and thetwo vertical walls. The nodes on the top surface are allowed to move freely. Thesolution for this simple problem is v = 0 an hydrostatic distribution of the pressurewhich is independent of the fluid viscosity. The problem is solved with a 2× 10× 10mesh of 3-node triangles.

Figure 82a shows the pressure distribution obtained by all methods. A convergedsolution which approximates practically the exact hydrostatic distribution is foundwith the PLS method in just two iterations.


00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951

00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.95 1

−2000

0

2000

4000

6000

8000

10000

xy

P

(a)

00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951

00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.95 1

−1000

0

1000

2000

3000

4000

5000

6000

xy

P

(b)

Figure 82: (a) Hydrostatic flow problem for a single fluid in square domain. (b) Two-fluidhydrostatic problem in square domain. Pressure distribution for both problemsobtained with PLS (with and without BI), GLS (with and without BI) and OSS(using T and Td). Results for all methods coincide.

0 1 2 3 4 5 6 7 8 9 1010

−5

10−4

10−3

10−2

10−1

100

101

Tot

al E

rror

Iteration

PLS : Nonlinear Convergence

Figure 83: Two-fluid hydrostatic problem. Convergence of the PLS method (with and withoutBI).

6.15.2 Two-fluid hydrostatic problem in a square domain

The same square container of the previous example is considered assuming that theupper half is filled with a liquid of density ρ = 10−3 Kg/m3. The viscosity is the samefor both fluids with µ = 10−3 Ns/m−2. The body forces, the boundary conditions andthe mesh are the same as for the previous example. The exact analytical solution isv = 0 in the whole container and a linear distribution of the pressure ranging from


p = 0 at the top (x2 = 1.0 m) to p = 10−2 Pa at x2 = 0.5 m; and again a lineardistribution of the pressure from p = 10−2 Pa at x2 = 0.5 m to p = 5000 Pa at x2 = 0.

Results for the pressure distribution are shown in Figure 82b. Numerical results forall methods studied coincide.

The converged solution for the PLS method is obtained in three iterations. The con-vergence history is shown in Figure 83.

0 1 2 3 4 5 60

0.5

1

1.5

2

Figure 84: Trapezoidal discretized by a symmetrical mesh of 2× 10× 10 3-node triangles.

6.15.3 Poiseuille flow in a trapezoidal domain

A trapezoidal domain Ω is considered with corner nodes given by: (0,0),(6,0),(4,2) and(2,2). The domain is discretized with a mesh of 2× 10× 10 3-node triangles (Figure84). A parabolic profile for the horizontal velocity is prescribed at both the inlet andoutlet lines.

Figure 85a shows the pressure distribution obtained with PLS method (with andwithout BI), the GLS method (with BI) and the OSS method (using T and Td). Resultsfor all these methods coincide. Figure 85b shows the GLS results not including the BIterm. Note the inaccuracies in the pressure distribution near the edge.

The same trend is observed for the results of the distribution of the v1 velocityshown in Figure 86. Note the slight increase in accuracy obtained in the PLS methodby including the BI term.

The convergence of the PLS solution with and without taking into account theBI terms is shown in Figure 87. The convergence improves when the BI terms areincluded (3 iterations versus 5 iterations).

6.15.4 Lid driven cavity problem

The flow in a driven square cavity of 1× 1 m2 is studied.The horizontal velocity on the top surface nodes has been prescribed to vp1 (x1, 1) = 1

m/s. The vertical velocity has also been prescribed to zero at all nodes on the topsurface with the exception of the central node with coordinates (0.5,1) which is leftfree to move in the vertical direction. The normal velocity at the bottom line and thetwo vertical walls has been prescribed to zero. The physical properties are ρ = 10−10

Kg/m3, g = 0 N/m2, µ = 1 Ns/m2.It can be easily verified that, for the material properties chosen, the value of the

stabilization parameter τi for the PLS method is approximately constant over thewhole analysis domain and equal to

τi = τ 'h2i16µ

=10−2

16= 6.25× 10−3m

3s

Kg(6.81)


00.30.60.91.21.51.82.12.42.7 33.33.63.94.24.54.85.15.45.7 6 0 0.10.20.30.40.50.60.70.80.911.11.21.31.41.51.61.71.81.92−5

−4

−3

−2

−1

0

1

y

x

P

(a)

00.30.60.91.21.51.82.12.42.7 33.33.63.94.24.54.85.15.45.7 6 0 0.10.20.30.40.50.60.70.80.911.11.21.31.41.51.61.71.81.92−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

y

xP

(b)

Figure 85: Poiseuille flow in a trapezoidal domain. (a) Pressure distribution obtained withPLS (with and without BI), GLS (with BI) and OSS (using T and Td). (b) Pressuredistribution obtained with GLS without BI

Figure 88 shows the pressure distributions in the cavity for all the methods studiedusing a 2× 20× 20 mesh of 3-node triangles. An analysis of the results of Figure 88

shows that (a) the PLS method captures better the singularity of the pressure valuesat the top corner nodes, (b) the effect of the BI terms is irrelevant for the PLS methodbut has a positive influence in the GLS method in terms of a better capture of thepressure singularity, (c) the diagonal form of matrix T introduce a slight diffusion inthe OSS results. The pressure contour lines for the considered methods is shown inFigure 89.

Figure 90 shows the convergence of the PLS results. The convergence curve is prac-tically the same accounting or not for the BI terms. Convergence is slower in this casedue to the pressure singularity at the top corners.

6.15.5 Manufactured flow problem in a trapezoidal domain

The trapezoidal domain of Figure 84 is discretized by a series of symmetrical meshconsisting of 2×n×n 3-node triangular elements and using n ∈ 10, 12, 14, 16, 18, 20,24, 32, 36, 40. The following relative error norms are used to study the convergencerates of the considered methods

ehvH1 =‖v − v‖1‖v‖1

=

√∫Ω∇(v − v) : ∇(v − v) dΩ√∫

Ω∇v : ∇v dΩ(6.82a)

ehpL2 =‖p− p‖0‖p‖0

=

√∫Ω(p− p)2 dΩ√∫

Ω p2 dΩ

(6.82b)


00.30.60.91.21.51.82.12.42.7 33.33.63.94.24.54.85.15.45.7 6 0 0.10.20.30.40.50.60.70.80.911.11.21.31.41.51.61.71.81.920

0.1

0.2

0.3

0.4

0.5

0.6

0.7

y

x

Ux

(a) PLS (without BI)

00.30.60.91.21.51.82.12.42.7 33.33.63.94.24.54.85.15.45.7 6 0 0.10.20.30.40.50.60.70.80.911.11.21.31.41.51.61.71.81.920

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

y

x

Ux

(b) GLS (without BI)

00.30.60.91.21.51.82.12.42.7 33.33.63.94.24.54.85.15.45.7 6 0 0.10.20.30.40.50.60.70.80.911.11.21.31.41.51.61.71.81.920

0.1

0.2

0.3

0.4

0.5

0.6

0.7

y

x

Ux

(c) PLS (with BI)

00.30.60.91.21.51.82.12.42.7 33.33.63.94.24.54.85.15.45.7 6 0 0.10.20.30.40.50.60.70.80.911.11.21.31.41.51.61.71.81.920

0.1

0.2

0.3

0.4

0.5

0.6

0.7

y

x

Ux

(d) GLS (with BI) and OSS (with T and Td)

Figure 86: Poiseuille flow in a trapezoidal domain. Results for the velocity v1 (denoted ux inthe figure obtained with (a) PLS (without BI), (b) GLS (without BI), (c) PLS (withBI), (d) GLS (with BI) and OSS (with T and Td)

where, (v, p) is the finite element approximation of the exact solution (v,p). Thenumerical solutions corresponding to the GLS, PLS, PLS+π and OSS methods, arecompared with the following solutions: the nodally exact interpolant denoted by(Ihv, Ihp) and the best approximation (BA) with respect to the L2 norm (for pres-sure) and the H1 semi-norm (for velocity) denoted by P0hp and P1hv respectively. Thesolutions Ihv, Ihp, P0hp and P1hv can be found as shown in Eq.(6.83).


0 1 2 3 4 5 6 7 8 9 1010

−5

10−4

10−3

10−2

10−1

100

101

Tot

al E

rror

Iteration

PLS: Convergence without BI term

(a)

0 1 2 3 4 5 6 7 8 9 1010

−5

10−4

10−3

10−2

10−1

100

101

Tot

al E

rror

Iteration

PLS: Convergence with BI term

(b)

Figure 87: Poiseuille flow in a trapezoidal domain. Convergence of the PLS solution. (a) With-out BI term. (b) With BI terms

Ihv := Nava; Ihp := Napa (6.83a)∫Ω

qh(p− P0hp) dΩ = 0 ∀ qh ∈ Qh

⇒ ‖ p− P0hp ‖0 6 ‖ p− p ‖0 ∀ p ∈ Qh(6.83b)

∫Ω

∇vh : ∇(v − P1hv) dΩ = 0 ∀ vh ∈ Vh

⇒ ‖ v − P1hv ‖1 6 ‖ v − v ‖1 ∀ v ∈ Vh(6.83c)

where, (va,pa) represent the nodal values of the exact solution (v,p), Qh ⊂ L2(Ω)

and Vh ⊂ H10(Ω). In the current example Qh and Vh are the solution spaces spannedby the 3-node triangle shape functions.

Consider a manufactured flow problem in which choosing the force term f =

(f1, f2), with f1 = µ(6x− 17), f2 = 0, we have the exact solution to the Stokes problemas v = (v1, v2), with v1 = y(2 − y)/2, v2 = 0 and p = µ(3x2 − 18x+ 1). Note thatchanging the magnitude of µ, any inf-sup stable Galerkin-FEM will give numericalsolutions that would scale proportional to the exact solution. Thus, for this manufac-tured problem, the relative errors eh

vH1and eh

pL2will be independent of µ.

The ehvH1

and ehpL2

convergence rates for the GLS, PLS, PLS+π and OSS methodsusing µ = 1 are shown in Figure 91.

Figure 91a illustrates the ehvH1

error lines for the considered methods. The error lineof the GLS method is slightly shifted above the error lines of Ihv (label: ‘Interpolant’)and P1hv (label: ‘H1-BA’). The error line of the OSS method shows a slight deviationfrom those of Ihv and P1hv, but it quickly merges with the later error lines on meshrefinement. The error line of the PLS method shows an improvement over the GLSmethod on coarse meshes but this advantage is lost on finer meshes wherein the two


error lines merge. The error line of the PLS+π method practically coincides with theerror lines of Ihv and P1hv.

Figure 91b illustrates the ehpL2

error lines for the considered methods. The error lineof the GLS method not only shows the greatest deviation from the error lines of Ihp(label: ‘Interpolant’) and P0hp (label: ‘L2-BA’) but also a sub-optimal convergence rate.The OSS method on the other hand show a convergence rate similar to that of Ihpand P0hp and the improvement over the GLS method is clear. The PLS and PLS+πmethods show convergence rates similar to that of the GLS and the OSS methodsrespectively. Nevertheless, the error lines of the former two methods are located closerto the error lines of Ihp and P0hp showing improved accuracy in the considered norms.The increase in accuracy is particularly relevant for the PLS+π method as clearly seenin Figure 91b.

The PLS results for each of the meshes studied were obtained in some 10 iterationswhereas typically 7 iterations were needed to obtain each of the PLS+π solutions.

6.16 conclusions

We have presented a family of stabilized finite element (FE) methods derived viafirst and second order the finite calculus (FIC) procedures. We have shown that sev-eral well known existing stabilized FE methods such as the penalty technique, theGalerkin Least Square (GLS) method, the Pressure Gradient Projection (PGP) methodand the Orthogonal Sub-Scales (OSS) method are recovered from the general residual-based FIC stabilized form. New stabilized FE methods such as the Pressure LaplacianStabilization (PLS) and the PLS+π method with consistent nonlinear forms of the sta-bilization parameters have been derived. The distinct feature of the PLS and PLS+πmethods is that the stabilization terms depend on the discrete residuals of the mo-mentum and the incompressibility equations.

The numerical results obtained for the Stokes problems solved in this work showthat the PLS method and, in particular, the PLS+π method provide accurate solutionsthat improve in several cases the results of the traditional GLS and OSS methods. Re-sults presented in [158] indicate that the PLS methods has also a superior performancethan the PGP method for some problems.

The prize to be paid for this increase in accuracy is the higher computational costassociated to the nonlinear solution which is intrinsic to the PLS method. The poten-tial and advantages of the PLS method will be clearer for transient problems solvedvia staggered schemes or for nonlinear flow problems which invariably require aniterative scheme.


0 0.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951

00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951−2000

−1500

−1000

−500

0

500

1000

1500

2000

xy

P


0 0.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951

00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951−2000

−1500

−1000

−500

0

500

1000

1500

2000

xy

P(b) PLS (with BI)

0 0.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951

00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951−800

−600

−400

−200

0

200

400

600

800

xy

P

(c) GLS (without BI)

0 0.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951

00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951−1000

−800

−600

−400

−200

0

200

400

600

800

1000

xy

P

(d) GLS (with BI)

0 0.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951

00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951−1500

−1000

−500

0

500

1000

1500

xy

P

(e) OSS (with Td)

0 0.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951

00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951−1500

−1000

−500

0

500

1000

1500

xy

P

(f) OSS (with T)

Figure 88: Pressure elevation plots for the lid driven cavity problem.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

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0.5

0.6

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2

−2−2

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−2

00

00

00

0

0 0

2

22

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22

4

44

4

4

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4

4

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66

6

6 6

6

6

8

88

8

8

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3030

40

40

4040

50

50

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60

60

70

70

70

80

80

80

100

100y

x

Pressure Contour

1822.0153−1822.0153


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

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0.9

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−100

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−6

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2−

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2

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00

00

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0

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2

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22

2

2

2

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44

4

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66

6

6

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88

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1010

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4040

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5050

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60

60

70

70

70

80

80

80

100

100

y

x

Pressure Contour

1841.053−1841.053

(b) PLS (with BI)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

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−100

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2

−2−2

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00

00

00

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44

4

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66

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88

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10

10

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2020

30

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30

30

40

40

4040

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50

50

60

60

60

70

70

70

80

80

100

100

y

x

Pressure Contour

786.0623−786.0623

(c) GLS (without BI)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

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100

100

y

x

Pressure Contour

905.8695−905.8695

(d) GLS (with BI)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

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0.3

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2

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60

70

70

70

80

80

80

100

100

y

x

Pressure Contour

1119.3583−1119.3583

(e) OSS (with Td)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

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00

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60 60

70

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80

80

80

100

100

y

x

Pressure Contour

1268.6564−1268.6564

(f) OSS (with T)

Figure 89: Pressure contour plots for the lid driven cavity problem.


0 1 2 3 4 5 6 7 8 9 10

10−4

10−3

10−2

10−1

100

101

Tot

al E

rror

Iteration

PLS : Nonlinear Convergence

Figure 90: Lid driven cavity problem. Convergence of PLS results. Convergence curve with orwithout BI terms are similar

−0.85 −0.8 −0.75 −0.7 −0.65 −0.6 −0.55 −0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2−1.65

−1.6

−1.55

−1.5

−1.45

−1.4

−1.35

−1.3

−1.25

−1.2

−1.15

−1.1

−1.05

−1

−0.95

−0.9

−0.85

log(h)

eh vH

1

(v − v) relative error in H1 semi-norm

GLSOSSPLSPLS+πInterpolant

H1−BA

(a)

−0.85 −0.8 −0.75 −0.7 −0.65 −0.6 −0.55 −0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2−4

−3.75

−3.5

−3.25

−3

−2.75

−2.5

−2.25

−2

−1.75

−1.5

−1.25

−1

log(h)

eh pL

2

(p − p) relative error in L2 norm

GLSOSSPLSPLS+πInterpolant

L2−BA

(b)

Figure 91: Convergence rates for the GLS, OSS, PLS and PLS+πmethods: (a) the relative errorsin the velocity (eh

vH1) and (b) the relative errors in the pressure (eh

pL2)

I have fought the good fight, I have finished the race, I have kept the faith.

— 2 Timothy 4:7.

C O N C L U S I O N S

In this monograph, we have proposed the following new stabilized finite element (FE)based Petrov–Galerkin (PG) methods:

1. A high-resolution Petrov–Galerkin (HRPG) method for the singularly perturbedconvection–diffusion–reaction (CDR) problem. The prefix high-resolution is usedhere in the sense popularized by Harten [82]—second-order accuracy for smoothregimes and good shock-capturing in non-regular regimes.

2. A PG method that reproduces on structured meshes the alpha-interpolation ofthe Galerkin finite element method (FEM) and the classical central finite dif-ference method (FDM). Particularly for the Helmholtz problem, this method iscapable of providing numerical solutions with a higher-order dispersion accu-racy.

3. A FIC-based pressure Laplacian stabilization (PLS) method for the Stokes prob-lem.

In the following, we describe the salient features of the work presented in the re-spective chapters. We refer to the individual chapter conclusions, viz. §1.7, §2.6, §3.6,§4.5, §5.7 and §6.16 for a more detailed summary of the same.

Chapter 1: To fix ideas prior to the development of the HRPG method, we have donea detailed analysis of a consistency recovery method for the 1d convection–diffusionequation stabilized using the SUPG [22, 92] or the FIC [141] method. The residualcorrection is done by including a convective projection variable into the stabilizationterm (motivated by the OSS [34] method). No gain in the dispersion accuracy is foundby this procedure. For the steady-state case, the optimal expression of the stabilizationparameter on uniform meshes is derived. An ad hoc extension of this stabilizationparameter to non-uniform meshes failed to preclude the occurrence of weak node-to-node oscillations. Further, the discrete system obtained on uniform meshes usingthis optimal parameter is neither a matrix of positive-type nor a monotone matrix.Nevertheless, it verifies the necessary and sufficient condition given in [186] for adiscrete maximum principle to hold. Unfortunately, all the conditions on the discretesystem, except for it being a matrix of positive-type, are difficult to identify a priori.This poses a strategical difficulty in the design of shock-capturing methods and hence,this consistency recovery procedure is not preferred in the development of the HRPGmethod.

Chapter 2: The weak form associated with the HRPG method consists of the stan-dard Galerkin terms, a linear upwinding term and a nonlinear shock-capturing term.The structure of the HRPG method in 1d (except for the definition of the stabilizationparameters) is identical to that of the consistent approximate upwind (CAU) Petrov–Galerkin method [68]. The distinction is that in multi-dimensions the upwinding is

189

190 conclusions

not streamline and the shock-capturing term is neither isotropic nor purely crosswind.Dropping the linear upwinding term, the expression in 1d of the stabilization param-eter β multiplying the shock-capturing term is found by relating it with the diffusionintroduced by the discrete-upwinding operation [120] on the Galerkin terms. It waspointed out earlier by Idelsohn et al. [99] that the transient term can be modeled asan instantaneous reaction term. Further, it was pointed out earlier by Codina [30] thatthe linear upwinding term can be interpreted as to contribute additional convection(negative upwind direction) and diffusion (rank one tensor) effects. Using these ideasthe effective convection, diffusion and reaction coefficients (for the transient problemand using the linear upwinding term) are calculated. Thus, for the transient caseand/or including the linear upwinding term, it is these effective coefficients that areused in the expression for β derived earlier. For the steady-state case, β depends onlyon the problem data, whereas for the transient case a nonlinear dependence existswhich guarantees the independence of the steady-state solution on the time step used.Several 1d examples are presented that illustrate the good treatment of the global,Gibbs and dispersive oscillations that otherwise plague the numerical solution of thesingularly perturbed CDR problem.

Chapter 3: A multi-dimensional extension of the HRPG method using multi-linearblock FEs is presented. First, we design a nondimensional element number that quan-tifies the characteristic layers which are found only in higher dimensions. This is doneby matching the width of the characteristic layers to the width of the parabolic lay-ers found for a fictitious 1d reaction–diffusion problem. The nondimensional elementnumber is then defined using this fictitious reaction coefficient, the diffusion coeffi-cient and an appropriate element size. Next, we introduce anisotropic element lengthvectors li and the stabilization parameters αi,βi calculated along these li. Except forthe modification to include the new dimensionless number that quantifies the charac-teristic layers, the definition of αi,βi are a direct extension of their counterparts in 1d.Using αi,βi and li, objective characteristic tensors associated with the HRPG methodare defined. The numerical artifacts across the characteristic layers are manifested asthe Gibbs phenomenon. Hence, we treat them just like the artifacts formed across theparabolic layers in the reaction-dominant case. Several 2d examples are presented thatsupport the design objective—stabilization with high-resolution.

Chapter 4: We present a simple domain-based higher-order compact scheme involv-ing two parameters α1,α2 on structured meshes for the Helmholtz equation. Makingthe parameters equal, we obtain the linear interpolation of the stencils obtained usingthe Galerkin FEM and the classical FDM. The stencil obtained by taking the parame-ters as distinct is denoted as the ‘nonstandard compact stencil’. In this case, the dif-fusion and production terms obtained from the Galerkin FEM and the classical FDMstencils are interpolated distinctly. Generic expressions for the parameters are giventhat guarantee a dispersion accuracy of sixth-order should the parameters be distinctand fourth-order should they be equal. In the later case, an expression for the param-eter is given that minimizes the maximum relative phase error of the scheme. Manyexisting higher-order compact schemes proposed within an algebraic setting can berecovered from the current scheme, viz. the alpha-interpolation method [110, 140](α1 = 0, α2 = α), the Vichnevetsky and Bowles scheme [187] (α1 = α, α2 = 1), thefourth-order generalized Padé approximation studied in [81, 168] (α1 = α2 = 0.5) andthe sixth-order quasi-stabilized FEM [10]. Recall that it is impossible to circumvent the

conclusions 191

pollution effect associated with the Helmholtz equation (hence one can only proposequasi-stabilized methods) and that the maximum dispersion accuracy attainable oncompact meshes is of sixth-order [10]. We remark that the proposed nonstandard com-pact stencil has an additional structure that reduces its abstractness. This additionalstructure (alternate point-of-view) is the key to develop PG methods which naturallyextend these schemes to unstructured meshes.

Chapter 5: The basic idea is to design test functions whose inner product with thestandard FE shape functions result in the lumped mass matrix. These test functionsare designed to have the following features: a) to be piecewise polynomials of thesame degree as the FE shape functions, b) to be a partition of unity (only in the sensethat they add up to unity) and c) to have a compact support. The last condition allowsus to construct test function spaces that vanish at the Dirichlet boundaries and thus,advocating its admittance into weak formulations. However, this condition makesthese test functions discontinuous at the element boundaries. As the row lumpingtechnique is a critical step in the design of these test functions (to fulfill the partition ofunity constraint), the current work is restricted only to those FEs where this techniquemakes sense—simplicial FEs and multi-linear block FEs. We show that using these testfunctions with an appropriate single-valued model on the element boundaries, it ispossible to recover the classical FDM stencil of the Helmholtz equation on structuredmeshes. The linear interpolation on the element boundaries (specified by α1) and theelement interiors (specified by α2) of these test functions with the standard FE shapefunctions, will result in a new class of test functions. These new test functions definethe proposed PG method involving two parameters α1,α2 that yields the nonstandardcompact stencil of the Helmholtz equation on structured meshes. Making α1 = α2 werecover the linear interpolation of the Galerkin FEM and the classical FDM stencilson structured meshes. The proposed PG method provides the counterparts of thesetwo schemes on unstructured meshes and allows the treatment of natural boundaryconditions (Neumann or Robin) and the source terms in a straight-forward manner.The additional cost of implementation of the proposed PG method is just the evalua-tion of the element boundary integrals. All the algebraic evaluations are done at theelement level unlike the QOPG method [128] where it is done at the patch level. Thisfeature allows the proposed PG method to be easily incorporated within an ‘assemble-by-elements’ data structure. The choice of the parameters α1 = α2 = (1/2) render theproposed PG method independent of the problem and mesh data. In this sense andfor this choice, the proposed PG method could be labeled ‘parameter-free’.

Chapter 6: New stabilized FE methods such as the pressure Laplacian stabilization(PLS) method and the PLS+π method with consistent nonlinear forms of the stabi-lization parameters are derived for the Stokes problem using the finite calculus (FIC)procedures. The distinct feature of these methods is that the stabilization terms de-pend on the discrete residuals of the momentum and the incompressibility equations.The PLS and PLS+π methods also introduce a boundary stabilization term. A similarboundary integral modification for the GLS method was proposed in [56]. Therein, itwas shown that accounting for this term is relevant when lower-order FEs are used.On the contrary, residual correction techniques as proposed in [111] are required. Un-like for the GLS method, the adverse effect of excluding the boundary stabilizationterm is found to be small for the PLS method. We believe this is due to the nonlinearfine-tuning of the stabilization terms in the PLS method. The performance of the PLS

192 conclusions

method is enhanced by adding to the momentum residual that appears in the stabi-lization terms its fine-scale projected counterpart (motivated by the OSS method) andthus, resulting in the PLS+π method. The numerical results obtained for the Stokesproblems solved in this work show that the PLS method and, in particular, the PLS+πmethod provide accurate solutions that improve in several cases the results of the tra-ditional GLS and OSS methods. The prize to be paid for this increase in accuracy isthe higher computational cost associated to the nonlinear solution which is intrinsicto the PLS method. The potential and advantages of the PLS method will be clearerfor transient problems solved via staggered schemes or for nonlinear flow problemswhich invariably require an iterative scheme.

outlook

In the following we list some aspects that are not addressed in this monograph whichnaturally constitute the future lines of work.

1. Simplicial FEs are not considered in the multi-dimensional extension of theHRPG method for the CDR problem. The primary obstacle in this line is toidentify anisotropic element length vectors and the subsequent definition of thecharacteristic tensors that not only guarantee the objectivity of the method butalso yield numerical solutions with a crisp layer resolution. The current pro-posal when applied to simplicial FEs violates the objectivity with respect to theadmissible node numbering permutations.

2. For the new PG method proposed for the Helmholtz problem, it will be interest-ing to study the dispersion accuracy of stencils made up of a symmetric patchof simplicial elements. For instance, patches that have a honeycomb structuremight yield stencils with higher-order dispersion accuracy. Following the ap-proach taken for bilinear block FEs, it is possible to provide different models forthe PG weights on the element boundaries. The benefits if any of this idea willbe explored in future works. Finally, the application of the new PG method toengineering problems involving time-harmonic wave propagation phenomenonwill also be explored.

3. For the convection–diffusion problem, the semi-discrete dispersion plots corre-sponding to the FIC/SUPG method (see Figure 1) reveals a possible gain in thedispersion accuracy by using an alpha-interpolated mass matrix in the transientterms. This is unlike what is observed for the Galerkin and the FIC_RC/OSSmethods, where the best dispersion accuracy was obtained using the consistentmass matrix. Such an alpha-interpolated mass matrix can be attained in a vari-ationally consistent manner using the PG formulation presented in chapter 5.Further studies will be done for the CDR problem exploring the benefits of us-ing the test functions presented in chapter 5 to treat the unsolved issues relatedto the HRPG method—Gibbs phenomenon of the solution derivatives.

4. Study the Navier-Stokes equations wherein the momentum equation and the in-compressibility constraint are stabilized using the HRPG and the PLS methods,respectively. Clearly, these two methods fall under the umbrella of FIC-basedmethods, of course with more elaborate stabilization terms.

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This monograph was typeset with LATEX 2ε using André Miede’s style package avail-able via CTAN as “classicthesis”. The package uses Hermann Zapf’s Palatino andEuler type faces (Type 1 PostScript fonts URW Palladio L and FPL were used). The typo-graphic style follows Bringhurst’s suggestions as presented in his book—The Elementsof Typographic Style.

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stabilized finite element methods for convection-diffusion

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